Fast Twitch Formula
Q: Does This Program Provide Every Set And Rep?
A: Absolutely! We take all the guesswork out of the training for the Athlete so they can focus on what matters…getting better!
Q: How Long Do These Workouts Take? Will They Fit My Schedule?
They take 15-25 minutes and fit PERFECTLY with other training programs, or busy 3 Sport Athlete Schedules!
Q: Do You Need Any Equipment For The Fast Twitch Formula?
A: Nope! Everything is bodyweight! You just need a step, or something you can find around the house that is elevated.
Q: How Do I Know My Kids Will Use This Training?
A: We built these workouts to give an instant feedback loop to young Athletes (very similar to what you’d see in a Video Game). Each exercise they perform within their 20-30 minute training sessions, they will be writing down their score. Through training thousands of young Athletes, this alone brings out a different competitive edge. You will be in awe when you see your Children competing at levels you never thought they had in them.
Q: Will I Lose Access To My Account?
A: NOPE! You will receive LIFETIME ACCESS when you order during this sale!
Q: What If I Play Multiple Sports?
A: That is EXACTLY why we built this training…for “The Busy Athlete”. We know how jam packed a Student Athletes schedule is so that’s why we built the workouts to be as lean and efficient as possible. There is not a single wasted rep!
Q: Does This Work For All Sports?
A: Yes! Regardless of the Sport, Speed And Agility Matters! Speed translates to more than just running…Power, Strength, and even Confidence on the Field or Court!
Q: What If I Don’t Like The Training?
A: Do not worry! That is why we offer a 60 Day Money Back Guarantee! If at any time you are not satisfied with your order, or you do not buy into the training, then we will refund you anytime over the next 60 days!
Q: Can I mix “The Fast Twitch Formula” with other training programs or Practice and Games?
A: Absolutely! We built this Workout with exactly that in mind! We wanted to make something that A) Kids buy into that they enjoy doing and competing in. B) Something that works perfectly around their busy schedules (only 20-30 minutes 3 times per week), and C) Something that gives them instant feedback by having them track their scores on every single exercise!
Q: What If I Am More Advanced And I Already Use Weights? Is This Still For Me?
A: We created this Workout for anyone who wants to get better footwork, get quicker, and more agile with just 20-30 minutes of time. This training works perfectly in-season, or even simultaneously while training with other programs (like the 30 day TTS System for example). Every exercise is tracked and measured so as you progress through the workouts you will be able to see if you are getting quicker or not. These Workouts are also seamless to use “In Season” so you do not overdo it.
Q.Are Your Workouts Digital Or Will They Be Shipped To My House?
A.All Twice The Speed workouts are digital and you will have immediate access to them. There is NOTHING that is shipped to your house.
Q.I Am An International Customer, Can I Get These Workouts?
A.Yes, because everything is digital, you will get access to the workouts INSTANTLY!
Q: What If I Have The TTS Resistance Bands, Will I Need Those For These Workouts?
A: You will not need them for the “Fast Twitch Formula”, but we do add bonus training that include Resistance Bands exercises because we have 100,000+ Customers who have them. We are throwing them in on the house for this Special Sale!
Fast Twitch Formula — | Twice The Speed Store
Q.Do You Offer A Money Back Guarantee On Your Programs?
A.YES! All of our workouts have at least a 60 day money back guarantee!
Q.What Forms Of Payments Do You Accept?
A.We accept ALL forms of payment. Visa, Mastercard, Amex, Paypal and you information is 100% secure.
Q.I Am An International Customer, Can I Get These Workouts?
A.Yes, because everything is digital, you will get access to the workouts INSTANTLY!
Q.
Can Advanced Athletes Do These Workouts?
A.Absolutely! Our workouts will work directly with how advanced you are. The more advanced of an athlete, the harder the workouts will be (and vice versa).
Q.Can I Do Multiple Programs Together At Once?
A.Yes we actually encourage you to integrate multiple programs of ours into your weekly regimen. We do NOT recommend you do more than 3 speed training, or other high intensity training sessions per week. The other days we would recommend that you are working on your upper body strength and explosiveness.
Q.What Sport Does TTS Help The Most?
A.These workouts are built for athletes who are looking to be the fastest, and most explosive they can possibly be. You will run faster, and feel much lighter on your feet which in return will make you dominate your sport.
Q.How Old Do You Have To Be For TTS Workouts?
A.Our youngest athlete who did our program was 6 years old, and our oldest was in his mid 60’s. We show multiple levels of intensity to suit your needs.
Q.Do I Need A Weight Room?
A.Although we do recommend a weight room for more advanced athletes, our workouts can be done (and we show you) with no weights in the comfort of your own home.
Q.Are Your Workouts Digital Or Will They Be Shipped To My House?
A.All Twice The Speed workouts are digital and you will have immediate access to them. There is NOTHING that is shipped to your house.
Test Your Fast-Twitch Muscles | ACTIVE
When it comes to building a better body, every guy is looking for an edge. And while some men might opt for a ‘roid trip to an underground pharmacy, the rest of us want a safer, smarter shortcut to more muscle. And I’ve found your advantage: fast-twitch muscle training. It’s the X factor that’ll help you pack on new muscle, add strength, and even burn more fat. But before I reveal the secret, let’s make one thing clear: Nothing can help you increase the quantity of your fast-twitch fibers. That was determined at birth. This leaves you with a choice: Pray that you won the genetic lottery, or find the best way to make your fast-twitch fibers bigger. Follow this two-step approach, and you’ll build more muscle than than you ever thought possible.
Test Your Fast-Twitch Fibers
You can activate your fast-twitch fibers two ways—by lifting heavier weights or by lifting lighter weights very quickly. Take this test to determine your fast-twitch ratio. The result will tell you how you need to lift in order to see the fastest improvement.
Step 1: Test your 1-rep max on the bench press (See below). Then rest five minutes.
Step 2: Select a weight that’s 45 percent of your 1-rep max. (So if your max is 225 pounds, you’ll start with about 100 pounds.) Try to perform five reps in five seconds.
Step 3: If you succeed, rest 1 to 2 minutes and then repeat the test, this time using 5 to 10 percent more weight. Keep adding 5 to 10 percent until you can no longer complete five reps in five seconds.
Step 4: Calculate your fast-twitch ratio: Simply divide the heaviest weight you could lift in five seconds by your 1-rep max. If you lifted 135 pounds in five seconds and your max is 225, your ratio would be 60 percent.
5-rep test/1-rep max
= fast-twitch ratio
How to Test Your 1-Rep Max
Using a spotter, perform a barbell bench press. Start with half of your estimated 1-rep max, or 1RM (the amount of weight you think you can press only once). Do five or six reps with perfect technique. Now add 10 percent more weight but subtract one rep. Rest two minutes. Repeat this pattern until you do one rep with about 90 percent of your estimated 1RM. Rest three to five minutes, and try your estimated max. If you achieve it, then that’s your true 1RM. If you fail, then use the 90 percent weight; if it’s too easy, add 10 percent to your estimated 1RM.
THE WORKOUT
Now that you’ve determined your fast-twitch ratio, select one of the workouts below to do as your upper-body routine at least twice a week, making sure you never do an upper-body routine two days in a row. Alternate between exercises that share the same number (1A and 1B, for example) until you complete all exercises in the pairing. Then move on to the next exercise. Select a weight that allows you to perform at least the minimum number of reps listed.
Extraordinary fast-twitch fiber abundance in elite weightlifters
Abstract
Human skeletal muscle fibers exist across a continuum of slow → fast-twitch. The amount of each fiber type (FT) influences muscle performance but remains largely unexplored in elite athletes, particularly from strength/power sports. To address this nescience, vastus lateralis (VL) biopsies were performed on World/Olympic (female, n = 6, “WCF”) and National-caliber (female, n = 9, “NCF”; and male, n = 6, “NCM”) American weightlifters. Participant accolades included 3 Olympic Games, 19 World Championships, 25 National records, and >170 National/International medals. Samples were analyzed for myosin heavy chain (MHC) content via SDS-PAGE using two distinct techniques: single fiber (SF) distribution (%) and homogenate (HG) composition. The main finding was that these athletes displayed the highest pure MHC IIa concentrations ever reported in healthy VL (23±9% I, 5±3% I/IIa, 67±13% IIa, and 6±10% IIa/IIx), with WCF expressing a notable 71±17% (NCF = 67±8%, NCM = 63±16%). No pure MHC IIx were found with SF. Secondary analysis revealed the heavyweights accounted for 91% of the MHC IIa/IIx fibers, which caused a correlation between this FT and body mass. Additionally, when compared to SF, HG overestimated MHC I (23±9 vs. 31±9%) and IIx (0±0 vs. 3±6%) by misclassifying I/IIa fibers as I and IIa/IIx fibers as IIx, highlighting the limitation of HG as a measure of isoform distribution. These results collectively suggest that athlete caliber (World vs. National) and/or years competing in the sport determine FT% more than sex, particularly for MHC IIa. The extreme fast-twitch myofiber abundance likely explains how elite weightlifters generate high forces in rapid time-frames.
Citation: Serrano N, Colenso-Semple LM, Lazauskus KK, Siu JW, Bagley JR, Lockie RG, et al. (2019) Extraordinary fast-twitch fiber abundance in elite weightlifters. PLoS ONE 14(3):
e0207975.
https://doi.org/10.1371/journal.pone.0207975
Editor: Nir Eynon, Victoria University, AUSTRALIA
Received: November 7, 2018; Accepted: February 15, 2019; Published: March 27, 2019
Copyright: © 2019 Serrano et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: Funding for this project was provided by Renaissance Periodization. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Italian physician Stefano Lorenzini made the first distinction of “red” and “white” muscle fibers (myofibers) in 1678, and almost 200 years later (1873) French histologist Louis-Antoine Ranvier confirmed the existence of two distinct myofiber types in vertebrate skeletal muscle. Reintroduction of the skeletal muscle biopsy procedure in 1962 [1] allowed scientists to begin exploring the topic in athletes and resulted in the discovery that each FT is comprised of a unique MHC isoform signature. Human skeletal muscle therefore contains three pure (MHC I, IIa, and IIx) and several hybrid (single myofibers that co-expresses multiple MHC isoforms) FTs [2]. The pure and hybrid FTs combine to form a robust slow → fast continuum (MHC I → I/IIa → IIa → IIa/IIx → IIx) with each displaying specific morphological, metabolic, and contractile properties [3–6]. FT% (the relative quantity of each FT in a given muscle) influences whole muscle function [7] and is often highly correlated with athletic performance [3, 7–13].
Extensive evidence indicates endurance athletes possess a slow-twitch myofiber majority [9, 10, 12, 14, 15], yet comparatively, far few investigations have explored FT in speed, power, or strength athletes. Initial research in the 1970–80’s found that resistance-trained men expressed high quantities (~60–65%) of fast-twitch fibers [11, 12, 15, 16], which was substantiated by later studies on elite powerlifters [17] and national-caliber (‘Olympic’) weightlifters [8]. This work provided an important foundation, but used sub-elite participants [18] and/or laboratory methods that failed to accurately resolve the highly prevalent hybrids [19–21]—which compromises measurement fidelity and produces erroneous FT% conclusions [19, 22–25]. More precise techniques were developed in the early 1990’s that allowed proper quantification of FT% by analyzing each SF.
Since this time only 13 studies (Table 1) implemented SF in young speed, power, or strength-trained individuals [5, 13, 19, 20, 22, 23, 25–30], and only 3 included females (n = 13, total). Only 5/13 included athletes: unknown-caliber male sprinters (n = 6) [25], male soccer players (n = 8) [24], elite female track and field athletes from a combination of pole vault, heptathlon, 100 and 400 m hurdles, and long jump events (n = 6) [20], National-caliber male bodybuilders (n = 8) [19], and a former World-champion male sprinter (n = 1) [13]. Accurately accounting for the full FT spectrum resulted in all five studies finding far lower MHC IIa concentrations than expected (52%, 30%, 16%, 39%, and 34%, respectively). The extremely low 16% found by Parcell et al. (2003) [20] is possibly explained by sex as females are often purported to possess more slow-twitch fibers than men [31, 32]. Such sex-specific phenotypes are often the case in murine models [31], but the topic remains unexplored in athletes. Moreover, these data are difficult to interpret as the athletes sampled were from a combination of several dissimilar events.
Numerous other knowledge gaps persist because in over 50 years of human muscle FT research only two studies have utilized SF with elite (i. e., world or international) athletes (one male sprinter and six female track and field) and no research has done so with any strength or power athletes. Thus, the purpose of this study was to examine the FT% of elite weightlifters to provide novel insight into the phenotype of competitive female and male strength and power athletes.
Methods
Experimental approach to the problem
Twenty-one elite (‘Olympic’) Weightlifters (15 female, 6 male) underwent resting VL biopsies between 2–96 hours after competing in either the International Weightlifting Federation World Championships or the USA Weightlifting American Open Finals (2017). All procedures and risks were explained to the athletes prior to obtaining written consent and completing medical and exercise history questionnaires. Performance records (taken from this event) in the snatch and clean and jerk (1RM), competition medals, and other accolades were gathered from personal interviews and publically available records from these or other sanctioned meets. Each muscle sample was analyzed for MHC content using two distinct FT techniques: SF and HG. The California State University, Fullerton Institutional Review Board approved all experimental procedures prior to any testing and consent was received in oral and written format.
Participants
Participants were subdivided into three categories; WCF (n = 6 female), NCF (n = 9), and NCM (n = 6). Athletes were considered “World-caliber” if they were on the most recent Olympic or World team and competed at the most recent national event. Athletes were considered “National-caliber” if they were top 5 placers at the 2017 American Open Finals meet but had never been on a World or Olympic team. Athletes spanned multiple weight categories, had a minimum of two years of national competition experience, had competed exclusively for the United States of America, and were otherwise eligible for all American national meets (Table 2). Athlete accolades at the time of data collection included participation in 3 Olympic Games, 19 World Championships, 11 Pan American Championships, 49 National Championships, 32 American Opens, 8 University National Championships, and 25 Junior World/Pan American/National Championships. Participants also held 25 national records and >170 national/international medals either at the time of the study or in the past. One athlete had tested positive for substances prohibited by the World Anti-Doping Agency and was suspended from the sport for two years prior to participating in the study.
Procedures
Muscle biopsies.
Following 30 minutes of supine rest, athletes underwent a mid-muscle belly (approximately halfway between the greater trochanter and patella) biopsy of the VL. A detailed description of the biopsy procedure has been previously described by our lab [9, 22, 23, 33]. Briefly, a small area of the thigh was numbed by injection of a local anesthetic (Xylocaine/Lidocaine without epinephrine). An approximately ¼ inch incision was made in the superficial cutaneous tissues. Muscle samples were obtained using the Bergström technique with suction [1], immediately cleansed of excess blood and connective tissue, divided into approximately 10–15 mg strips, placed into cold skinning solution (125 mM K propionate, 2. 0 mM EGTA, 4.0 mM ATP, 1.0 mM MgCl_{2}, 20.0 mM imidazole [pH 7.0], and 50% [vol ml/vol ml] glycerol), and stored at -20° C for at least one week. Each sample was split such that a portion (~5 mg) could be used for SF or HG. The incision site was cleaned, pulled closed with a sterile Band-Aid, and covered with sterile gauze and cohesive bandage tape.
MHC FT identification.
All biopsy samples were analyzed for MHC via SDS-PAGE using two distinct techniques: SF and HG. For SF, individual fibers (N = 2,147; 102 ± 3 fibers per athlete) were mechanically isolated with fine tweezers under a light microscope and placed in 80 μl of sodium dodecyl sulfate (SDS) buffer (1% SDS, 23 mM EDTA, 0.008% bromophenol blue, 15% glycerol, and 715 mM b-mercaptoethanol [pH 6.8]). HG samples (~5 mg) were hand homogenized and then diluted between 1:10 to 1:50 based on sample amount and protein quantity. As described in detail elsewhere [5, 9, 22, 23, 27], 1–2 μl aliquots of both SF or HG (run separately) were then loaded into individual wells in a 3. 5% loading and 5% separating gel (SDS-PAGE), run at 5°C for 15.5 hours (SE 600 Series; Hoefer, San Francisco, CA, USA), and silver stained for MHC identification. The SF approach used known molecular weights and standards to identify the MHC isoform (MHC I, I/IIa, IIa, IIa/IIx, and IIx) of each individual myofiber. This enabled the most accurate calculation of the FT% within the muscle sample [21]. HG utilized densitometry (ImageJ, National Institutes of Health, Bethesda, MD) to quantify the relative MHC protein composition (i.e., percent area occupied by each pure isoform; MHC I, IIa, and IIx) of each sample, which is highly correlated with FT area [34]. Thus, SF indicates how frequently each isoform exists but cannot address how much area each FT occupies within the muscle. HG addresses the latter, but cannot delineate hybrids, therefore inaccurately quantifying FT% [9, 21–25, 27].
Statistical analysis
Potential differences between groups in descriptive information were examined via ANOVA. For SF, potential differences in FT% between groups were assessed via a 3 (group: WCF, NCF, NCM) x 4 (fiber type: MHC I, I/IIa, IIa, IIa/IIx) ANOVA. For HG, potential differences in FT composition between groups were examined via a 3 (group: WCF, NCF, NCM) x 3 (fiber type: MHC I, IIa, IIx) ANOVA. Comparison of SF vs. HG was accomplished by a 2 (group: SF, HG) x 3 (fiber type: MHC I, IIa, IIx) ANOVA. Effect size was calculated with Cohen’s D (0.2 = small difference, 0.5 = medium difference, and 0.8 = large difference) to identify the magnitude of difference between two groups. Pearson Product Moment Correlations (r) were assessed for WCF, NCF, and NCM between 1RM, body mass, and SF FT%. All individual FT data are reported in Table 3. Data are reported as mean ± standard deviation (SD), unless otherwise noted. Significance was established a priori at an alpha level of p < 0.05. All analyses were performed with SPSS (SPSS Statistics Version 24, IBM).
Results
Descriptive
WCF were significantly older than NCF, but not NCM (Table 2). WCF also had significantly more years of sport competition experience than NCF and NCM. NCM exceeded both WCF and NCF in relative strength in both the snatch 1RM and clean and jerk 1RM.
SF distribution
FT% for all lifters combined was 23 ± 9% I, 5 ± 3% I/IIa, 67 ± 13% IIa, and 6 ± 10% IIa/IIx. No MHC IIx or I/IIa/IIx fibers were identified. No significant differences existed between groups, despite WCF possessing 8% (absolute, not percent difference) less MHC I than NCF (d = 0.88) and NCM (d = 0.78) (Fig 1). The difference in MHC IIa between WCF and NCM (also 8%) was also not statistically significant, but had a moderate effect size (d = 0.50). The vast majority of the MHC IIa/IIx fibers (91%) belonged to just five lifters, all of whom competed in the heavyweight or super heavyweight categories (≥ 90 kg for women and ≥105 kg for men). This produced significant correlations between body mass and MHC IIa/IIx frequency for WCF (r = 0. 919, p = 0.010) and NCF (r = 0.826, p = 0.006) and a trend for NCM (r = 0.757, p = 0.080).
HG composition
FT composition for all lifters combined was 31 ± 9% I, 67 ± 9% IIa, and 3 ± 6% IIx. MHC I tended (p = 0.08) to be lower in WCF (24 ± 7%) than NCF (33 ± 9%, p = 0.125, d = 1.14) and NCM (35 ± 9%, p = 0.106, d = 1.33), yet MHC IIa was significantly higher (p = 0.046) in WCF (74 ± 6%) than NCM (61 ± 11%, p = 0.043, d = 1.39), but not NCF (65 ± 7%, p = 0.145, d = 1.28) and. FT was significantly different (p < 0.001) between SF and HG for MHC I (p = 0.005) and MHC IIx (p = 0.046), but not MHC IIa. SF MHC IIa/IIx and HG MHC IIx were highly correlated (r = 0.96, p < 0. 001). No correlations existed for SF or HG between FT% and snatch or clean and jerk relative 1RM when analyzed as subgroups or when combined together.
Discussion
The current study resulted in the most detailed investigation of muscle phenotype in Olympic and World-caliber anaerobic athletes published to date. These data are the first comparison of World vs. National-caliber athletes at the SF level. Additionally, they enabled the most precise description of FT% in strength or power sport competitors, and the first ever in females. The primary finding was that the pure MHC IIa abundance was the highest in healthy muscle (VL) ever reported, especially for females. This finding suggests athlete caliber and/or years competing in the sport influence FT% more than sex per se and also questions the pronouncement that male athletes possess more fast-twitch myofibers than females. Secondary findings revealed that our utilization of two different typing methods confirmed the limitations of HG for FT% (inappropriately categorizes MHC I/IIa as MHC I and MHC IIa/IIx as MHC IIx) and also allowed identification of a previously undocumented relationship between body mass and MHC IIa/IIx concentrations. The unique morphology and phenotypes in our participants highlight the need to further study elite anaerobic athletes, particularly females.
WCF contained the highest concentration of MHC IIa (71%) reported in the literature to our knowledge. NCF (67%) and NCM (63%) also possessed more MHC IIa than previous research in competitive male bodybuilders (40%) [16, 19] as well as male power/weightlifters [8, 11, 12, 15, 16, 18], elite female pole vaulters, heptathletes, 100 and 400 m hurdlers, and long jumpers (20), elite male hammer throwers [35], and resistance-trained men [18, 19, 22, 23, 27, 29, 36], which all ranged from 50–60%. Only six previous studies using SF have found pure MHC IIa concentrations of >50%, with just two reporting 60% (Table 1). The resulting minimal MHC I (~17–25%) in our athletes was strikingly lower than the previously described track and field athletes (57%) [20] and National-caliber bodybuilders (35%) [19]. These pronounced differences are likely explained by the substantial dissimilarities in training styles (e.g., external loading strategies, contraction type and velocity, training frequency, etc. ) between the various sports. More research is therefore needed to continue delineating the subtle but significant differences in FT% between athletes from a range of anaerobic sports and the specific role each training approach might play in altering MHC I and IIa distribution.
Although the differences in FT% between our three groups did not reach statistical significance, large effect sizes were evident and MHC IIa frequencies of 74%-89% occurred in 66% of WCF but only in 44% and 33% of NCF and NCM, respectively. Thus, scientists should further examine how FT% may separate World from National-level athletes as it would enhance our understanding of the physiological factors determining maximal human performance. For example, the only published report on a world-record holding anaerobic athlete found a FT profile remarkably different from our study or any other previous research in elite sprinters [13]. The minimal exploration in this area makes it difficult to determine if such a separation in FT profile between elite subgroups is a true and consistent phenomenon or merely an artifact of too little research.
Our groups differed in two other important characteristics; sex and years competing in the sport. Sex comparisons between athletes remain tenuous [31, 37] because nearly all investigations utilize non-gold standard FT% methods [21] and sedentary [38, 39] or “recreationally active” individuals [32]. Not only do our findings contradict the claim that women possess more slow-twitch myofibers than men [40], they illustrate the opposite when accounting for talent level (WCF < NCF = NCM). WCF had also been competing in the sport for ~5 years longer than both NCF and NCM. The current cross-sectional study-design precludes direct analysis, but extensive research affords strong support for exercise history as a critical determinate of FT% [2, 9, 26, 28, 30, 41–44]. Chronic exercise generally decreases hybrids [30, 42] and induces style-specific shifts in FT% such as increases in MHC I with endurance [9, 43, 45] or MHC IIa with sprint [28], plyometric [26], or strength training [36, 43, 44, 46]. For example, one study reported an increase in MHC IIa from 46% to 60% following 19 weeks of resistance training [36]. MHC IIa/IIx fibers appear particularly responsible for exercise-induced increases in MHC IIa and are thus uncommon in exercise-trained individuals [9, 20, 22–25, 29, 43]. A reduction of MHC IIx in favor of IIa following chronic resistance exercise is also purported extensively in the literature [28, 34], yet the overwhelming majority of this evidence comes from experiments with methodologies (e.g. HG) directly shown here and elsewhere [9, 24, 25, 47] to produce erroneous FT% conclusions.
Most research from the 1970’s– 2000’s utilized either ATPase histochemistry or HG SDS-PAGE to determine FT% [8, 14–17, 24, 25, 34, 36]. Similar to SF, histochemistry allows assessment of individual fibers for calculation of percent distribution, yet it does not enable simultaneous delineate of hybrids [36]. HG suffers the same drawback and actually indicates FT area/composition [34] more so than distribution making it greatly influenced by the size of each fiber; which is not uniform across all FTs (particularly in resistance trained individuals) [48]. All three approaches hold strong merit and are often correlated to each other [34, 49] and performance [8], but are clearly not interchangeable for maximally precise FT% assessment. In the current study, HG accurately quantified MHC IIa (within 0–4%), but not I or IIx. MHC I was overestimated by 8% percent (23 vs. 31%), which is largely explained by the non-differentiated MHC I/IIa fibers (5%). HG also greatly exaggerated MHC IIx, particularly in individuals with >4% MHC IIa/IIx. The inability of HG to account for MHC IIa/IIx explains why MHC IIx appear common in some studies [50] even though the actual abundance of pure MHC IIx fibers in healthy human skeletal muscle is extraordinarily rare; typically <0.1% [9, 22–25, 27] and 0 of the >2,100 isolated fibers from the current sample. Thus, the seeming conversion of MHC IIx to IIa with exercise is more precisely IIa/IIx changing to IIa.
MHC IIa/IIx hybrids are typically inversely associated with muscle health and physical activity [2, 9, 30, 43, 47, 51–54]. Yet, the heavyweights (male and female) expressed irregularly high concentrations (24%) and accounted for 91% of all MHC IIa/IIx myofibers, explaining the correlations between body mass and MHC IIa/IIx quantities. Terzis and colleagues (2010) noted a similar abnormal abundance of MHC IIx (typed via HG, so likely IIa/IIx) in six large (116 kg, body fat composition >22%), but presumably highly strength-trained throwers [35]. Body composition was not assessed in the current study and little research exists on well-trained, but high body mass individuals. Not knowing the amount of muscle vs. fat on these larger participants limits the ability to speculate on potential mechanisms. Thus, additional studies with a larger sample size across a broader spectrum of physical size are required to truly interpret the correlations between body mass and MHC IIa/IIx prevalence and to explore possible mechanisms.
Another juxtaposition was that of FT% and performance. Previous work in 94 kg male competitive weightlifters found strong correlations between FT composition (via HG) and the percentage of total area in a muscle that each FT occupies to both snatch 1RM and vertical jump height [8], but not clean and jerk 1RM. We failed to identify any such correlations (when all subjects were combined or sub-grouped), but also utilized multiple sexes and weight classes. Thus, while FT% differed between our groups, that factor alone did not predict performance among our lifters. Several possible explanations exist for this discrepancy. First, FT area may determine whole muscle strength more than FT%. Second, neither studies found correlations to the clean and jerk, which is heavier and slower than the snatch or vertical jump. This compliments previous isokinetic research [23] and indicates FT% does not predict performance on strength tasks among strength-trained individuals. FT% probably determines movement speed more than force production [7]. Further speculation on this point is unwarranted as limitations prohibited the ability to assess FT-specific size or contractile properties, which likely differed significantly across our groups [55] and are known to changes with training [3, 48].
Conclusion
This study provides novel insight into the muscle phenotype of elite competitive strength and power athletes and highlights the need for more research in this area. The extreme fast-twitch abundance partially explains how elite weightlifters are able to generate high forces in short time-frames. Our data also indicate that athlete caliber and years competing in the sport dictate FT% more than sex per se, but more work is needed to draw firm conclusions as a single biopsy may not perfectly represent the entire muscle [56]. Most athletes contained few hybrids and no MHC IIx or I/IIa/IIx, except the heavyweights who possessed atypically high quantities of IIa/IIx. Future research should use high fidelity techniques to explore FT-specific distribution, size, and contractile properties in female and male athletes of various caliber, sports, and body size; ideally across several years of competition. The resulting data could have practical significance if it enabled experimentation of differing training volumes or recovery protocols based on athlete-specific FT properties [57]. Scientifically, our findings importantly contribute to the knowledge-base of fiber type-specific physiology.
Acknowledgments
The authors would like to thank Irene S. Tobias and Cameron Yen for their help with this project.
References
- 1.
Bergstrom J. Muscle electrolytes in man determined by neutron activation analysis on needle biopsy specimens. The Scandinavian Journal of Clinical and Laboratory Investigations. 1962;14 (Suppl. 68). - 2.
Pette D, Staron RS. Myosin isoforms, muscle fiber types, and transitions. Microsc Res Tech. 2000;50(6):500–9. Epub 2000/09/22. pmid:10998639. - 3.
Methenitis S, Karandreas N, Spengos K, Zaras N, Stasinaki AN, Terzis G. Muscle Fiber Conduction Velocity, Muscle Fiber Composition, and Power Performance. Med Sci Sports Exerc. 2016;48(9):1761–71. Epub 2016/04/30. pmid:27128672. - 4.
Galpin AJ, Raue U, Jemiolo B, Trappe TA, Harber MP, Minchev K, et al. Human skeletal muscle fiber type specific protein content. Anal Biochem. 2012;425(2):175–82. Epub 2012/04/04. pmid:22469996; PubMed Central PMCID: PMCPMC3358799. - 5.
Tobias IS, Lazauskas KK, Arevalo JA, Bagley JR, Brown LE, Galpin AJ. Fiber type-specific analysis of AMPK isoforms in human skeletal muscle: advancement in methods via capillary nano-immunoassay. J Appl Physiol (1985). 2017. Epub 2018/01/24. pmid:29357518. - 6.
Bagley JR. Fibre type-specific hypertrophy mechanisms in human skeletal muscle: potential role of myonuclear addition. J Physiol. 2014;592(23):5147–8. Epub 2014/12/03. 592/23/5147 [pii]. pmid:25448185; PubMed Central PMCID: PMC4262329. - 7.
Bottinelli R, Pellegrino MA, Canepari M, Rossi R, Reggiani C. Specific contributions of various muscle fibre types to human muscle performance: an in vitro study. J Electromyogr Kinesiol. 1999;9(2):87–95. Epub 1999/03/31. S1050-6411(98)00040-6 [pii]. pmid:10098709. - 8.
Fry AC, Schilling BK, Staron RS, Hagerman FC, Hikida RS, Thrush JT. Muscle fiber characteristics and performance correlates of male Olympic-style weightlifters. J Strength Cond Res. 2003;17(4):746–54. Epub 2003/12/12. pmid:14666943. - 9.
Bathgate KE, Bagley JR, Jo E, Talmadge RJ, Tobias IS, Brown LE, et al. Muscle health and performance in monozygotic twins with 30 years of discordant exercise habits. Eur J Appl Physiol. 2018. Epub 2018/07/15. pmid:30006671. - 10.
Costill DL, Daniels J, Evans W, Fink W, Krahenbuhl G, Saltin B. Skeletal muscle enzymes and fiber composition in male and female track athletes. J Appl Physiol. 1976;40(2):149–54. Epub 1976/02/01. pmid:129449. - 11.
Tesch PA, Karlsson J. Muscle fiber types and size in trained and untrained muscles of elite athletes. J Appl Physiol (1985). 1985;59(6):1716–20. Epub 1985/12/01. pmid:4077779. - 12.
Prince FP, Hikida RS, Hagerman FC. Human muscle fiber types in power lifters, distance runners and untrained subjects. Pflugers Arch. 1976;363(1):19–26. Epub 1976/05/06. pmid:131933. - 13.
Trappe S, Luden N, Minchev K, Raue U, Jemiolo B, Trappe TA. Skeletal muscle signature of a champion sprint runner. J Appl Physiol (1985). 2015;118(12):1460–6. Epub 2015/03/10. japplphysiol.00037.2015 [pii]. pmid:25749440; PubMed Central PMCID: PMC4469925. - 14.
Gollnick PD, Armstrong RB, Saubert CWt, Piehl K, Saltin B. Enzyme activity and fiber composition in skeletal muscle of untrained and trained men. J Appl Physiol. 1972;33(3):312–9. Epub 1972/09/01. pmid:4403464. - 15.
Tesch PA, Thorsson A, Kaiser P. Muscle capillary supply and fiber type characteristics in weight and power lifters. J Appl Physiol Respir Environ Exerc Physiol. 1984;56(1):35–8. Epub 1984/01/01. pmid:6693333. - 16.
Tesch PA, Larsson L. Muscle hypertrophy in bodybuilders. Eur J Appl Physiol Occup Physiol. 1982;49(3):301–6. Epub 1982/01/01. pmid:6890445. - 17.
Fry AC, Webber JM, Weiss LW, Harber MP, Vaczi M, Pattison NA. Muscle fiber characteristics of competitive power lifters. J Strength Cond Res. 2003;17(2):402–10. Epub 2003/05/14. pmid:12741885. - 18.
Galpin AJ, Fry AC, Chiu LZ, Thomason DB, Schilling BK. High-power resistance exercise induces MAPK phosphorylation in weightlifting trained men. Appl Physiol Nutr Metab. 2012;37(1):80–7. Epub 2012/01/10. pmid:22220922. - 19.
Kesidis N, Metaxas TI, Vrabas IS, Stefanidis P, Vamvakoudis E, Christoulas K, et al. Myosin heavy chain isoform distribution in single fibres of bodybuilders. Eur J Appl Physiol. 2008;103(5):579–83. Epub 2008/05/08. pmid:18461351. - 20.
Parcell AC, Sawyer RD, Craig Poole R. Single muscle fiber myosin heavy chain distribution in elite female track athletes. Med Sci Sports Exerc. 2003;35(3):434–8. Epub 2003/03/06. pmid:12618572. - 21.
Pandorf CE, Caiozzo VJ, Haddad F, Baldwin KM. A rationale for SDS-PAGE of MHC isoforms as a gold standard for determining contractile phenotype. J Appl Physiol (1985). 2010;108(1):222–2; author reply 6. Epub 2010/01/08. pmid:20054086. - 22.
Arevalo JA, Lynn SK, Bagley JR, Brown LE, Costa PB, Galpin AJ. Lower-Limb Dominance, Performance, and Fiber Type in Resistance-trained Men. Med Sci Sports Exerc. 2017. Epub 2017/12/23. pmid:29271846. - 23.
Bagley JR, McLeland KA, Arevalo JA, Brown LE, Coburn JW, Galpin AJ. Skeletal Muscle Fatigability and Myosin Heavy Chain Fiber Type in Resistance Trained Men. J Strength Cond Res. 2017;31(3):602–7. pmid:27984439. - 24.
Andersen JL, Klitgaard H, Bangsbo J, Saltin B. Myosin heavy chain isoforms in single fibres from m. vastus lateralis of soccer players: effects of strength-training. Acta Physiol Scand. 1994;150(1):21–6. Epub 1994/01/01. pmid:8135120. - 25.
Andersen JL, Klitgaard H, Saltin B. Myosin heavy chain isoforms in single fibres from m. vastus lateralis of sprinters: influence of training. Acta Physiol Scand. 1994;151(2):135–42. Epub 1994/06/01. pmid:7942047. - 26.
Malisoux L, Francaux M, Nielens H, Renard P, Lebacq J, Theisen D. Calcium sensitivity of human single muscle fibers following plyometric training. Med Sci Sports Exerc. 2006;38(11):1901–8. Epub 2006/11/11. pmid:17095922. - 27.
Murach KA, Bagley JR, McLeland KA, Arevalo JA, Ciccone AB, Malyszek KK, et al. Improving human skeletal muscle myosin heavy chain fiber typing efficiency. J Muscle Res Cell Motil. 2016;37(1–2):1–5. pmid:26842420. - 28.
Parcell AC, Sawyer RD, Drummond MJ, O’Neil B, Miller N, Woolstenhulme MT. Single-fiber MHC polymorphic expression is unaffected by sprint cycle training. Med Sci Sports Exerc. 2005;37(7):1133–7. Epub 2005/07/15. pmid:16015129. - 29.
Raue U, Terpstra B, Williamson DL, Gallagher PM, Trappe SW. Effects of short-term concentric vs. eccentric resistance training on single muscle fiber MHC distribution in humans. Int J Sports Med. 2005;26(5):339–43. Epub 2005/05/17. pmid:15895315. - 30.
Williamson DL, Gallagher PM, Carroll CC, Raue U, Trappe SW. Reduction in hybrid single muscle fiber proportions with resistance training in humans. J Appl Physiol (1985). 2001;91(5):1955–61. pmid:11641330. - 31.
Haizlip KM, Harrison BC, Leinwand LA. Sex-based differences in skeletal muscle kinetics and fiber-type composition. Physiology (Bethesda). 2015;30(1):30–9. Epub 2015/01/07. pmid:25559153; PubMed Central PMCID: PMCPMC4285578. - 32.
Norman B, Esbjornsson M, Rundqvist H, Osterlund T, von Walden F, Tesch PA. Strength, power, fiber types, and mRNA expression in trained men and women with different ACTN3 R577X genotypes. J Appl Physiol (1985). 2009;106(3):959–65. Epub 2009/01/20. 91435.2008 [pii]. pmid:19150855. - 33.
Bagley JR, Galpin AJ. Three-dimensional printing of human skeletal muscle cells: An interdisciplinary approach for studying biological systems. Biochem Mol Biol Educ. 2015;43(6):403–7. pmid:26345697. - 34.
Fry AC, Allemeier CA, Staron RS. Correlation between percentage fiber type area and myosin heavy chain content in human skeletal muscle. Eur J Appl Physiol Occup Physiol. 1994;68(3):246–51. Epub 1994/01/01. pmid:8039521. - 35.
Terzis G, Spengos K, Kavouras S, Manta P, Georgiadis G. Muscle fibre type composition and body composition in hammer throwers. J Sports Sci Med. 2010;9(1):104–9. Epub 2010/01/01. pmid:24149393; PubMed Central PMCID: PMC3737956. - 36.
Adams GR, Hather BM, Baldwin KM, Dudley GA. Skeletal muscle myosin heavy chain composition and resistance training. J Appl Physiol (1985). 1993;74(2):911–5. Epub 1993/02/01. pmid:8458814. - 37.
Prince FP, Hikida RS, Hagerman FC. Muscle fiber types in women athletes and non-athletes. Pflugers Arch. 1977;371(1–2):161–5. Epub 1977/10/19. pmid:145580. - 38.
Miller MS, Callahan DM, Tourville TW, Slauterbeck JR, Kaplan A, Fiske BR, et al. Moderate-intensity resistance exercise alters skeletal muscle molecular and cellular structure and function in inactive older adults with knee osteoarthritis. J Appl Physiol (1985). 2017;122(4):775–87. Epub 2017/01/14. pmid:28082334; PubMed Central PMCID: PMCPMC5407204. - 39.
Simoneau JA, Bouchard C. Human variation in skeletal muscle fiber-type proportion and enzyme activities. Am J Physiol. 1989;257(4 Pt 1):E567–72. Epub 1989/10/01. pmid:2529775. - 40.
Welle S, Tawil R, Thornton CA. Sex-related differences in gene expression in human skeletal muscle. PLoS One. 2008;3(1):e1385. Epub 2008/01/03. pmid:18167544; PubMed Central PMCID: PMCPMC2148100. - 41.
Pette D, Staron RS. Mammalian skeletal muscle fiber type transitions. Int Rev Cytol. 1997;170:143–223. Epub 1997/01/01. pmid:9002237. - 42.
Pette D. The adaptive potential of skeletal muscle fibers. Can J Appl Physiol. 2002;27(4):423–48. Epub 2002/11/22. pmid:12442355. - 43.
Bagley JR, Murach KA, Trappe S. Microgravity-Induced Fiber Type Shift in Human Skeletal Muscle Gravitational and Space Biology. 2012;26(1):34–40. - 44.
Liu Y, Schlumberger A, Wirth K, Schmidtbleicher D, Steinacker JM. Different effects on human skeletal myosin heavy chain isoform expression: strength vs. combination training. J Appl Physiol (1985). 2003;94(6):2282–8. Epub 2003/05/09. pmid:12736190. - 45.
Trappe S, Harber M, Creer A, Gallagher P, Slivka D, Minchev K, et al. Single muscle fiber adaptations with marathon training. J Appl Physiol (1985). 2006;101(3):721–7. Epub 2006/04/15. pmid:16614353. - 46.
Kryger AI, Andersen JL. Resistance training in the oldest old: consequences for muscle strength, fiber types, fiber size, and MHC isoforms. Scand J Med Sci Sports. 2007;17(4):422–30. Epub 2007/05/11. pmid:17490465. - 47.
Purves-Smith FM, Sgarioto N, Hepple RT. Fiber typing in aging muscle. Exerc Sport Sci Rev. 2014;42(2):45–52. Epub 2014/02/11. pmid:24508741. - 48.
Grgic J, Schoenfeld BJ. Are the Hypertrophic Adaptations to High and Low-Load Resistance Training Muscle Fiber Type Specific? Front Physiol. 2018;9:402. Epub 2018/05/04. pmid:29720946; PubMed Central PMCID: PMCPMC5915697. - 49.
Staron RS. Correlation between myofibrillar ATPase activity and myosin heavy chain composition in single human muscle fibers. Histochemistry. 1991;96(1):21–4. Epub 1991/01/01. pmid:1834618. - 50.
Aagaard P, Magnusson PS, Larsson B, Kjaer M, Krustrup P. Mechanical muscle function, morphology, and fiber type in lifelong trained elderly. Med Sci Sports Exerc. 2007;39(11):1989–96. Epub 2007/11/08. pmid:17986907. - 51.
Borina E, Pellegrino MA, D’Antona G, Bottinelli R. Myosin and actin content of human skeletal muscle fibers following 35 days bed rest. Scand J Med Sci Sports. 2010;20(1):65–73. Epub 2009/11/04. pmid:19883388. - 52.
Malisoux L, Jamart C, Delplace K, Nielens H, Francaux M, Theisen D. Effect of long-term muscle paralysis on human single fiber mechanics. J Appl Physiol (1985). 2007;102(1):340–9. Epub 2006/10/14. pmid:17038491. - 53.
Klitgaard H, Zhou M, Richter EA. Myosin heavy chain composition of single fibres from m. biceps brachii of male body builders. Acta Physiol Scand. 1990;140(2):175–80. Epub 1990/10/01. pmid:2148462. - 54.
Klitgaard H, Zhou M, Schiaffino S, Betto R, Salviati G, Saltin B. Ageing alters the myosin heavy chain composition of single fibres from human skeletal muscle. Acta Physiol Scand. 1990;140(1):55–62. Epub 1990/09/01. pmid:2275405. - 55.
Staron RS, Hagerman FC, Hikida RS, Murray TF, Hostler DP, Crill MT, et al. Fiber type composition of the vastus lateralis muscle of young men and women. J Histochem Cytochem. 2000;48(5):623–9. Epub 2000/04/18. pmid:10769046. - 56.
Lexell J, Henriksson-Larsen K, Sjostrom M. Distribution of different fibre types in human skeletal muscles. 2. A study of cross-sections of whole m. vastus lateralis. Acta Physiol Scand. 1983;117(1):115–22. Epub 1983/01/01. pmid:6858699. - 57.
Jones N, Kiely J, Suraci B, Collins DJ, de Lorenzo D, Pickering C, et al. A genetic-based algorithm for personalized resistance training. Biol Sport. 2016;33(2):117–26. Epub 2016/06/09. 1198210 [pii]. pmid:27274104; PubMed Central PMCID: PMC4885623.
Fast Twitch Workout Plan
Post Your Comments?
Fast Twitch Muscle Workouts – Where HIIT Meets Strength
5 hours ago “The HIIT workout was better at providing that necessary stimulus to the muscles to have a more favorable training adaptation,” lead researcher …
Estimated Reading Time: 4 mins
Website: Squatwolf.com ^{}
Category: Use words in a sentence
Favorable
Your Guide to FastTwitch Muscle Training
1 hours ago Your fast–twitch muscle fibers, also known as Type II fibers, are the fibers responsible for explosive movements, things like vertical leaps, 40-yard …
Estimated Reading Time: 7 mins
Website: Menshealth.com ^{}
Category: Use to in a sentence
Fast, Fibers, For
Fasttwitch Muscle Exercises Using Resistance Bands
9 hours ago 3. Lateral Hops. Fast-twitch muscle workout. If you are looking for a workout you can do anywhere, these exercises are a must. Using a …
Estimated Reading Time: 8 mins
Website: Victoremgear.com ^{}
Category: Use words in a sentence
Fast, For
Becoming A Fast Twitch Machine Bodybuilding.com
4 hours ago Transformation of slow to fast or fast to slow through training influence. Transformable Fibers . If you were to look at a muscle biopsy youâ€™d see both red and white along with various shades of each. The white being pure fast twitch and the red being pure slow twitch. Think of eating chicken, the white meat (breast) is fast twitch.
Estimated Reading Time: 11 mins
Website: Bodybuilding.com ^{}
Category: Use words in a sentence
Fast, Fibers
Fast Twitch Muscle Fibers Training 101 Gym Junkies
7 hours ago This Is Your Fast Twitch Muscle Fiber Recruitment Workout. Most of the exercises used for this workout are compound lifts. Of course, you may have heard of them being called “big lifts” as well. These are the most beneficial for you to achieve fast twitch muscle fiber recruitment. But there will be some isolation exercises thrown in as well.
Reviews: 4
Estimated Reading Time: 9 mins
Website: Gymjunkies.com ^{}
Category: Use words in a sentence
Fast, Fiber, For
How to Target FastTwitch Muscle Fibers Bodybuilding.com
3 hours ago Fatigue Is Your Friend. Along with intensity, fatigue is the second surefire way to increase fast–twitch muscle fiber recruitment. Your body’s first impulse is to recruit slow-twitch fibers, but once you fatigue those fibers, it has to recruit fast–twitch fibers to do what is being asked of it. This is where how you train is more important than how your workout looks on paper.
Is Accessible For Free: True
Estimated Reading Time: 7 mins
Website: Bodybuilding.com ^{}
Category: Use to in a sentence
Fatigue, Friend, Fast, Fiber, First, Fibers
3 Tips for Building Fast Twitch Muscles AlterG
3 hours ago Regardless of how you choose to build fast–twitch muscles, remember that no workout regimen is exclusive of your diet and sleep regimen. Your ability to perform these workouts without injury, as well as to recover properly after taxing workouts, is just as important to building fast–twitch muscle fibers as the exercises themselves.
Website: Alterg.com ^{}
Category: Use for in a sentence
Fast, Fibers
How to Train SlowTwitch and Fast Muscle & Fitness
7 hours ago You can get jacked or be diesel even if you have slow-twitch or fast–twitch muscle fibers right now. For the uninitiated, your muscles are made up of thousands of individual muscle fibers . “Think of it as a ponytail,” says Dr. Andy Galpin, an exercise physiologist who has spent more than 20 years studying human performance.
Estimated Reading Time: 4 mins
Website: Muscleandfitness.com ^{}
Category: Use to in a sentence
Fast, Fibers, For
Fast Twitch TrainingFitness Training
3 hours ago Fast Twitch muscles are the largest, most powerful muscle fibers in your body and have the most potential for growth and development. Fast Twitch Training (FTT) is a full service, cutting-edge fitness training facility that is committed to helping people live a more FIT LIFE by developing healthy habits in a safe, non-intimidating, community based environment.
Website: Fasttwitchtraininginc.com ^{}
Category: Use words in a sentence
Fast, Fibers, For, Ftt, Full, Fitness, Facility, Fit
How to train your fasttwitch muscles Six Star Pro Nutrition
2 hours ago When creating a workout to target fast–twitch muscle fibers, add explosive movements like box jumps, medicine ball slams and kettlebell swings to your routine. Ensure that part of your training program focuses on heavier power exercises – like power cleans, push presses and squats – and lift very heavy (90% or more of your 1-rep max) as
Website: Sixstarpro.com ^{}
Category: Use to in a sentence
Fast, Fibers, Focuses
Senior Exercise and Fast Twitch Muscles Healthy Living
2 hours ago Senior Workouts for Fast-Twitch Muscle. You can develop your fast-twitch muscles at your age the same way you did in your youth — by strength-training and doing speed drills. Keeping a regular weight-lifting routine, specifically for your lower body and core, will help you retain muscle altogether, which is half the battle.
Website: Healthyliving.azcentral.com ^{}
Category: Use and in a sentence
For, Fast
How to Develop Fast Twitch Muscle Fiber
3 hours ago High school athletes want powerful fast–twitch muscle fiber to make the starting lineup.Olympians want fast-muscle fiber to win the gold. Everyone interested in dropping body fat, toning muscle, and reversing the effects of the middle-age somatopause, should have a fitness plan that includes developing fast–twitch muscle fiber.
Website: Howtobefit.com ^{}
Category: Use to in a sentence
Fast, Fiber, Fat, Fitness
What Are FastTwitch Muscle Fibers? How to Train Fast
6 hours ago Fast–twitch muscle fibers are one of two types of skeletal muscle fibers, in addition to slow-twitch muscle fibers. Fast–twitch muscles are responsible for high-intensity work like heavy lifting
Website: Health.com ^{}
Category: Use to in a sentence
Fast, Fibers, For
What Is The Fast Twitch Formula Workout Bundle?
5 hours ago The Fast Twitch Formula is a “BRAND NEW” tried and true workout system that has been proven for over a decade of real-world testing on hundreds of athletes! Each workout can 100% be completed at home without any additional or costly gym equipment purchases.
Website: Training.twicethespeed.com ^{}
Category: Use words in a sentence
Fast, Formula, For
FastTwitch Muscle Fiber Training – Recoup Fitness
7 hours ago 1) Speed work. Short repeat intervals – traditional interval workouts help recruit intermediate and fast-twitch muscle fibers. By being used together, these two fiber types learn to interact more efficiently. Sprint work – hill sprints and maximum effort sprints help recruit the maximum amount of fast-twitch muscle fibers.
Website: Recoupfitness.com ^{}
Category: Use words in a sentence
Fast, Fibers, Fiber
How to build more fasttwitch muscle Coach
1 hours ago Use King’s explosive workouts to build fast–twitch muscle and correct muscle imbalances. Do ten sets in each workout, starting with six to eight reps of each move. Rest five minutes between sets.
Website: Coachmag.co.uk ^{}
Category: Use to in a sentence
Fast, Five
How to Develop FastTwitch Leg Muscles Livestrong.com
1 hours ago ACE Fitness: Slow-Twitch vs. Fast–Twitch Muscle Fibers Journal of Strength and Conditioning Research: The effects of endurance, strength, and power training on muscle fiber type shifting. Strength and Conditioning Research: Muscle Fiber Type Research Gate: Training Fast Twitch Muscle Fibers: Why and How
Website: Livestrong.com ^{}
Category: Use to in a sentence
Fitness, Fast, Fibers, Fiber
50 Rep “FAST Twitch” Workout For All Athletes YouTube
3 hours ago http://twicethespeed.com/Jack Cascio from Twice The Speed shows you a 50 rep Fast Twitch Muscle Circuit workout. This is a workout that is geared to help yo
Website: Youtube.com ^{}
Category: Use words in a sentence
From, Fast
Combining SlowTwitch & FastTwitch Workouts Woman The
6 hours ago Endurance aerobic workouts, such as jogging or cycling at a moderate intensity, focus on slow-twitch fibers. Lifting weights, sprinting and circuit training stimulate your fast–twitch fibers. To design a workout that engages both muscle fibers, you need to combine the two. High-intensity intervals can be designed to target both muscle fibers.
Website: Woman.thenest.com ^{}
Category: Use words in a sentence
Focus, Fibers, Fast
Bodyweight exercises to stimulate fasttwitch muscle
9 hours ago So I want to achieve a slim-athletic physique through only doing body-weight exercises. As I understand it, after you do more than 12-15ish reps of an exercise per set, you’re no longer utilising your fast–twitch muscle fibers which are the ones that are more capable of hypertrophy and so you’ll mainly be developing endurance with little strength/size improvement (slow-twitch).
Website: Fitness.stackexchange.com ^{}
Category: Use to in a sentence
Fast, Fibers
Fast Twitch Muscles: About, Benefits, Exercises, Vs. Slow
3 hours ago Fast twitch muscles help with sudden bursts of energy involved in activities like sprinting and jumping. Slow twitch muscles are better for …
Website: Healthline.com ^{}
Category: Use words in a sentence
Fast, For
How to Build a Basketball Body stack
1 hours ago Your basketball workout plan should build the fast–twitch muscles you need to jump and sprint. Below is a sample basketball workout plan. Adjust the sets and reps up or down for age or skill level
Website: Stack.com ^{}
Category: Use to in a sentence
Fast, For
The Best Sprint Workouts to Get Faster, Build Muscle, and
8 hours ago 3x 10–12 sec. @ 85% effort (or fast but not all-out) at 8% incline 90-sec. recovery walk or jog; 5x 10–12 sec. @ 95% effort (as fast as you can possibly go) at 1% incline 90-sec. recovery walk
Website: Mensjournal.com ^{}
Category: Use to in a sentence
Fast
Training fast twitch muscle fibers Rowperfect UK
7 hours ago Training Fast Twitch Muscle Fibers: Why and How Ernest W. Maglischo 1970 Lazy Meadow Lane Prescott, AZ 86303 USA [email protected] Abstract. With the finding that short, intense sprints can improve aerobic capacity (Tabata, et al., 1996), there has been a huge increase in the number of experts who
Website: Rowperfect.co.uk ^{}
Category: Use words in a sentence
Fast, Fibers, Finding
Fast Food: Usain Bolt’s Diet and Workout Program Man of Many
6 hours ago Usain Bolt’s Workout Plan. You don’t get to be the fastest man in history by diet alone. That’s why Usain Bolt’s training routine is as consistent as it is methodical, and executed on a daily basis. Given his above-average height and weight, Bolt focuses on generating propulsiveness by strengthening his fast–twitch muscle fibres.
Website: Manofmany.com ^{}
Category: Use and in a sentence
Fastest, Focuses, Fast, Fibres
A 15Minute HIIT Treadmill Workout to Run Faster
3 hours ago Try This 30-60-10 Treadmill Workout to Run Faster. For this treadmill workout, you’ll be running 30-second sprints with 60 seconds of walking recovery and repeating that cycle 10 times. But don’t completely empty your tank on the first few rounds. Your sprints should start out at about an eight on a scale of one to 10.
Website: Livestrong.com ^{}
Category: Use to in a sentence
Faster, For, First, Few
FAST TWITCH OR SLOW TWITCH, WHICH IS BEST FOR …
3 hours ago SHOP NOW CLOTHING AND SUPPLEMENTS: http://www.Jaycutler.comE BOOK: https://www.jaycutler.clubFREE NEWSLETTER: https://jaymail.jaycutler.clubTRIFECTA (MEALS):
Website: Youtube.com ^{}
Category: Use words in a sentence
Slow Twitch and Fast Twitch Muscle Mass Fitness VIP
2 hours ago Fast twitch muscle fibers are classified as either type 2a or type 2b. These are the fibers that are involved during weight training. Type 2a fibers are always used in a higher rep range, greater than 12, and also are the first to come into play in the 6 to 12 rep range.
Website: Fitness-vip.com ^{}
Category: Use and in a sentence
Fast, Fibers, First
Kegel Exercises a step by step howto guide for women
6 hours ago work both your fast twitch and slow twitch muscle fibers To give your pelvic floor a full workout, there are two types of focused kegel exercises you could perform. Quick or Short Muscle Contractions (Fast Twitch Muscle Exercise) – The first exercise is called a …
Website: Nafc.org ^{}
Category: Use a in a sentence
Fast, Fibers, Floor, Full, Focused, First
How to Use a Treadmill for Slow & FastTwitch Muscle Workouts
3 hours ago Improving Your Fast–Twitch Muscles On A Treadmill. If you are an athlete who relies on speed or strength such as a sprinter, basketball player, short-distance swimmer, running back or powerlifter, you will want to focus your efforts on developing fast–twitch muscles on your treadmill. The kinds of treadmill workouts you should be focusing on are:
Website: Proformpromocodes.com ^{}
Category: Use to in a sentence
Fast, Focus, Focusing
Fitness Friday: Organizing a workout This is the Loop
3 hours ago The final segment of your workout involves fast, explosive movements. Golf is a fast–twitch muscle sport. The swing itself lasts less than 1.5 seconds, so it’s important to train your muscles to
Website: Golfdigest.com ^{}
Category: Use a in a sentence
Final, Fast
The Difference Between Slow Twitch And Fast Twitch Muscle
8 hours ago The fast–twitch: slow-twitch muscle fiber ratio is determined genetically. No amount of exercise or training will change this ratio, but it is possible to convert type IIx muscle fibers to type IIa or vice versa.
Website: Sushifitness.com ^{}
Category: Use words in a sentence
Fast, Fiber, Fibers
27 Fast twitch muscle ideas muscle, twitch, workout
5 hours ago The Triple Threat Metabolic Workout Posted on June 22, 2015 This workout (metabolic workout to be exact) is designed to target a few key areas: Explosiveness – build fast-twitch muscle fiber and lower body power Metabolism – increase metabolism and turn your body into a fat burning machine Strength – build lean muscle mass and improve
Website: Pinterest.ca ^{}
Category: Use words in a sentence
Few, Fast, Fiber, Fat
17 Baseball Workouts To Get You Ready For the Season
7 hours ago It’s about fine tuning your core, staying flexible, and conditioning your fast twitch muscles to be ready for action at the crack of the bat. I’m not a personal trainer, nor do I play one on the internet – so that’s why we’ve done some digging to come up …
Website: Fivetoolschool.com ^{}
Category: Use words in a sentence
Fine, Flexible, Fast, For
Increase Your FastTwitch Potential With Isometrics
6 hours ago In fact, fast-twitch muscle fibers can contract ten times faster than slow-twitch fibers. This is the kind of muscle that you really want to develop to train speed – and it’s likely that you’re doing it so, so wrong. How to Train Fast-Twitch Muscles. If you want to train fast-twitch muscles, you have to move fast, right? No, you don’t. In
Website: Breakingmuscle.com ^{}
Category: Use words in a sentence
Fact, Fast, Fibers, Faster
Rowing and FastTwitch Muscles? : Fitness
9 hours ago It seems to offer a whole body workout with minimal impacts on your body’s joints. My problem is, my body is comprised mostly of fast–twitch muscles, which is mainly from me lifting every day and my track experience/workouts. I do not want to lose any of those, but I understand I will need mostly slow-twitch for rowing.
Website: Reddit.com ^{}
Category: Use and in a sentence
Fast, From, For
Personal Trainer & Fitness Coach In Apollo Beach, Ruskin
8 hours ago With my formal education in exercise science and my understanding of both anatomy and strength training, I will develop a fitness program tailored to your individual needs that will accomplish your goals safely and in a timely manner. If you’d like to talk about your fitness goals call me at (813) 294-2836 or complete the online form. Read My Story
Website: Fasttwitchfp.com ^{}
Category: Use words in a sentence
Formal, Fitness, Form
Rethink Your Rep Range Jim Stoppani
6 hours ago The fast rep workouts also caused the men to burn 5% more calories at rest after the workout was over. So be sure to change up your rep speeds just like you change up other aspects of your workouts. Keep normal-speed reps at the foundation of your training program, using them the majority of the time.
Website: Jimstoppani.com ^{}
Category: Use words in a sentence
Fast, Foundation
The Hike Forever Fitness Plan: Age 3550 Backpacker
9 hours ago B: Muscles Fast–twitch muscle fiber–the type used in sprinting and power moves–decreases between 4 and 10 percent per decade beginning at age 40 in sedentary adults. But slow-twitch muscle–the type used in endurance training–remains strong. The fix: Build power for butt-busting ascents with hill workouts, advises Lynn Millar.
Website: Backpacker.com ^{}
Category: Use words in a sentence
Fast, Fiber, Fix, For
The Ultimate Push Up Workout: Get Ripped Quick Maxim Online
5 hours ago The clapping push up is known for being a feat of strength. It is often used in film or TV to show characters as impossibly strong or manly. Besides being good for flaunting, the clapping push up also trains your explosive power and fast–twitch muscles. To start the clapping push up, assume the starting position of the traditional push-up.
Website: Maximonline.com ^{}
Category: Use words in a sentence
For, Feat, Film, Flaunting, Fast
How to get Faster at Running? Speed Drills for Runners
6 hours ago Speed workouts to increase fast twitch muscles It’s important to know there is a VAST difference between sprint speed and improving your speed over 13.1 miles. Sprinters are going to throw their arms hugely up and down, while distance runners are focused on a smaller swing to conserve energy.
Website: Runtothefinish.com ^{}
Category: Use to in a sentence
Fast, Focused
Simply The Best Workout Programs You Can Find
7 hours ago Designed by fitness models Dave Dreas and Jeremy Scott this program gives you everything it takes and more to shed pounds and get ripped fast. The program comes with a 10-week training manual, training templates, a nutrition guide, grocery store guide and 10-week HIIT guide as well as a complete strength training video library and more.
Website: Theathleticbuild.com ^{}
Category: Use words in a sentence
Fitness, Fast
Lacrosse flexibility and performance Training for girls
7 hours ago Our lacrosse performance training program is designed for the lacrosse player who needs to get bigger, stronger, faster, quicker and more explosive. Our program is divided into two segments; strength training and speed training. The strength training program is designed to: Improve mobility and stability. Improve core strength.
Website: Perfect-performancenova.com ^{}
Category: Use and in a sentence
For, Faster
MOTOR UNITS AND MUSCLE TWITCHES
MOTOR UNITS
Image drawn by BYU-I student Nate Shoemaker Spring 2016
The motor neurons that innervate skeletal muscle fibers are called alpha motor neurons. As the alpha motor neuron enters a muscle, it divides into several branches, each innervating a muscle fiber (note this in the image above). One alpha motor neuron along with all of the muscle fibers it innervates is a motor unit . The size of the motor unit correlates with the function of the muscle. In muscles involved with fine, coordinated control, the motor units are very small with 3-5 muscle fibers per motor neuron. Muscles that control eye movement and muscles in our hands have relatively small motor units. On the other hand in muscles involved with more powerful but less coordinated actions, like the muscles of the legs and back, the motor units are large with 1000s of muscle fibers per motor neuron.
MUSCLE TWITCH
Title: File:1012 Muscle Twitch Myogram.jpg; Author: OpenStax College; Site:https://commons.wikimedia.org/wiki/File:1012_Muscle_Twitch_Myogram.jpg; License: This file is licensed under the Creative Commons Attribution 3.0 Unported license.
When an action potential travels down the motor neuron, it will result in a contraction of all of the muscle fibers associated with that motor neuron. The contraction generated by a single action potential is called a muscle twitch. A single muscle twitch has three components. The latent period, or lag phase, the contraction phase, and the relaxation phase. The latent period is a short delay (1-2 msec) from the time when the action potential reaches the muscle until tension can be observed in the muscle. This is the time required for calcium to diffuse out of the SR, bind to troponin, the movement of tropomyosin off of the active sites, formation of cross bridges, and taking up any slack that may be in the muscle. The contraction phase is when the muscle is generating tension and is associated with cycling of the cross bridges, and the relaxation phase is the time for the muscle to return to its normal length. The length of the twitch varies between different muscle types and could be as short as 10 ms (milliseconds) or as long as 100 ms (more on this later).
If a muscle twitch is just a single quick contraction followed immediately by relaxation, how do we explain the smooth continued movement of our muscles when they contract and move bones through a large range of motion? The answer lies in the ordering of the firing of the motor units. If all of the motor units fired simultaneously the entire muscle would quickly contract and relax, producing a very jerky movement. Instead, when a muscle contracts, motor units fire asynchronously, that is, one contracts and then a fraction of a second later another contracts before the first has time to relax and then another fires and so on. So, instead of a quick, jerky movement the whole muscle contraction is very smooth and controlled. Even when a muscle is at rest, there is random firing of motor units. This random firing is responsible for what is known as muscle tone. So, a muscle is never “completely” relaxed, even when asleep. However, if the neuron to a muscle is cut, there will be no “muscle tone” and this is called flaccid paralysis. There are several benefits of muscle tone: First it takes up the “slack” in the muscle so that when it is asked to contract, it can immediately begin to generate tension and move the limb. If you have ever towed a car you know what happens if you don’t take the slack out of the tow rope before starting to pull. The second thing muscle tone does is deter muscle atrophy.
TYPES OF MUSCLE CONTRACTION
Muscle contractions are described based on two variables: force (tension) and length (shortening). When the tension in a muscle increases without a corresponding change in length, the contraction is called an isometric contraction (iso = same, metric=length). Isometric contractions are important in maintaining posture or stabilizing a joint. On the other hand, if the muscle length changes while muscle tension remains relatively constant, then the contraction is called an isotonic contraction (tonic = tension). Furthermore, isotonic contractions can be classified based on how the length changes. If the muscle generates tension and the entire muscle shortens than it is a concentric contraction. An example would be curling a weight from your waist to your shoulder; the bicep muscle used for this motion would undergo a concentric contraction. In contrast, when lowering the weight from the shoulder to the waist the bicep would also be generating force but the muscle would be lengthening, this is an eccentric contraction. Eccentric contractions work to decelerate the movement at the joint. Additionally, eccentric contractions can generate more force than concentric contractions. Think about the large box you take down form the top shelf of your closet. You can lower it under total control using eccentric contractions but when you try to return it to the shelf using concentric contractions you cannot generate enough force to lift it back up. Strength training, involving both concentric and eccentric contractions, appears to increase muscle strength more than just concentric contractions alone. However, eccentric contractions cause more damage (tearing) to the muscle resulting in greater muscle soreness. If you have ever run downhill in a long race and then experienced the soreness in your quadriceps muscles the next day, you know what we are talking about.
Muscle size is determined by the number and size of the myofibrils, which in turn is determined by the amount of myofilament proteins. Thus, resistance training will induce a cascade of events that result in the production of more proteins. Often this is initiated by small, micro-tears in and around the the muscle fibers. If the tearing occurs at the myofibril level the muscle will respond by increasing the amount of proteins, thus strengthening and enlarging the muscle, a phenomenon called hypertrophy. This tearing is thought to account for the muscle soreness we experience after a workout. As mentioned above, the repair of these small tears results in enlargement of the muscle fibers but it also results in an increase in the amount of connective tissue in the muscle. When a person “bulks up” from weight training, a significant percent of the increase in size of the muscle is due to increases in the amount of connective tissue. It should be pointed out that endurance training does not result in a significant increase in muscle size but increases its ability to produce ATP aerobically.
FACTORS THAT INFLUENCE THE FORCE OF MUSCLE CONTRACTION
Obviously our muscles are capable of generating differing levels of force during whole muscle contraction. Some actions require much more force generation than others; think of picking up a pencil compared to picking up a bucket of water. The question becomes, how can different levels of force be generated?
Multiple-motor unit summation or recruitment: It was mentioned earlier that all of the motor units in a muscle usually don’t fire at the same time. One way to increase the amount of force generated is to increase the number of motor units that are firing at a given time. We say that more motor units are being recruited. The greater the load we are trying to move the more motor units that are activated. However, even when generating the maximum force possible, we are only able to use about 1/3 of our total motor units at one time. Normally they will fire asynchronously in an effort to generate maximum force and prevent the muscles from becoming fatigued. As fibers begin to fatigue they are replaced by others in order to maintain the force. There are times, however, when under extreme circumstances we are able to recruit even more motor units. You have heard stories of mothers lifting cars off of their children, this may not be totally fiction. Watch the following clip to see how amazing the human body can be. Muscle recruitment. (Video Transcription Available)
Title: 1013_Summation_Tetanus.jpg; Author: OpenStax; Site: http://cnx.org/contents/[email protected]:67/Anatomy-&-Physiology; License: This work is licensed by Rice University under a Creative Commons Attribution License License ( 3.0).
Wave summation: Recall that a muscle twitch can last up to 100 ms and that an action potential lasts only 1-2 ms. Also, with the muscle twitch, there is not refractory period so it can be re-stimulated at any time. If you were to stimulate a single motor unit with progressively higher frequencies of action potentials you would observe a gradual increase in the force generated by that muscle. This phenomenon is called wave summation. Eventually the frequency of action potentials would be so high that there would be no time for the muscle to relax between the successive stimuli and it would remain totally contracted, a condition called tetanus. Essentially, with the high frequency of action potentials there isn’t time to remove calcium from the cytosol. Maximal force, then, is generated with maximum recruitment and an action potential frequency sufficient to result in tetanus.
Title: 1011_Muscle_Length_and_Tension.jpg; Author: OpenStax; Site: http://cnx.org/contents/[email protected]:67/Anatomy-&-Physiology; License: This work is licensed by Rice University under a Creative Commons Attribution License License ( 3.0).
Initial Sarcomere Length: It has been demonstrated experimentally that the starting length of the sarcomere influences the amount of force the muscle can generate. This observation has to do with the overlap of the thick and thin filaments. If the starting sarcomere length is very short, the thick filaments will already be pushing up against the Z-disc and there is no possibility for further sarcomere shortening, and the muscle will be unable to generate as much force. On the other hand, if the muscle is stretched to the point where myosin heads can no longer contact the actin, then again, less force will be generated. Maximum force is generated when the muscle is stretched to the point that allows every myosin head to contact the actin and the sarcomere has the maximum distance to shorten. In other words, the thick filaments are at the very ends of the thin filaments. These data were generated experimentally using frog muscles that were dissected out and stretched between two rods. Intact muscles in our bodies are not normally stretched very far beyond their optimal length due to the arrangement of muscle attachments and joints.
However, you can do a little experiment that will help you see how force is lost when a muscle is in a very short or a very stretched position. This experiment will use the muscles that help you pinch the pad of your thumb to the pads of your fingers. These muscles are near maximal stretch when you extend your arm and also extend your wrist. As your wrist is cocked back into maximal extension, try to pinch your thumb to your fingers. See how weak it feels? Now, gradually flex your wrist back to a straight or neutral position. You should feel your pinch get stronger. Now, flex your elbow and your wrist. With your wrist in maximal flexion, the muscles you use to pinch with are near their most shortened position. Try pinching again. It should feel weak. But, again, as you extend your wrist back to neutral you should feel your pinch get stronger.
ENERGY SOURCE FOR MUSCLE CONTRACTION
The ultimate source of energy for muscle contraction is ATP. Recall that each cycle of a myosin head requires an ATP molecule. Multiply that by all of the myosin heads in a muscle and the number of cycles each head completes each twitch and you can start to see how much ATP is needed for muscle function. It is estimated that we burn approximately our entire body weight in ATP each day so it becomes apparent that we need to constantly replenish this important energy source. For muscle contraction, there are four ways that our muscles get the ATP required for contraction.
- Cytosolic ATP: This ATP represents the “floating” pool of ATP, or that which is present and available in the cytoplasm. This ATP requires no oxygen (anaerobic) to make it (because it is already there) and is immediately available but it is short lived. It provides enough energy for a few seconds of maximal activity in the muscle-not the best source for long term contraction. Nevertheless, for the muscles of the eyes that are constantly contracting quickly but for short periods of time, this is a great source.
- Creatine Phosphate: Once the cytosolic stores of ATP are depleted, the cell calls upon another rapid energy source, Creatine Phosphate. Creatine phosphate is a high energy compound that can rapidly transfer its phosphate to a molecule of ADP to quickly replenish ATP without the use of oxygen. This transfer requires the enzyme creatine kinase, an enzyme that is located on the M-line of the sarcomere. Creatine phosphate can replenish the ATP pool several times, enough to extend muscle contraction up to about 10 seconds. Creatine Phosphate is the most widely used supplement by weight lifters. Although some benefits have been demonstrated, most are very small and limited to highly selective activities.
- Glycolysis: Glycolysis, as the name implies, is the breakdown of glucose. The primary source of glucose for this process is from glycogen that is stored in the muscle. Glycolysis can function in the absence of oxygen and as such, is the major source of ATP production during anaerobic activity. This series of chemical reactions will be a major focus in the next unit. Although glycolysis is very quick and can supply energy for intensive muscular activity, it can only be sustained for about a minute before the muscles begin to fatigue.
- Aerobic or Oxidative Respiration: The mechanisms listed above can supply ATP for maybe a little over a minute before fatigue sets in. Obviously, we engage in muscle activity that lasts much longer than a minute (things like walking or jogging or riding a bicycle). These activities require a constant supply of ATP. When continuous supplies of ATP are required, the cells employ metabolic mechanisms housed in the mitochondria that utilize oxygen. We normally refer to these processes as aerobic metabolism or oxidative metabolism. Using these aerobic processes, the mitochondria can supply sufficient ATP to power the muscle cells for hours. The down side of aerobic metabolism is that it is slower than anaerobic mechanisms and is not fast enough for intense activity. However, for moderate levels of activity, it works great. Although glucose can also be utilized in aerobic metabolism, the nutrient of choice is fatty acids. As described below, slow-twitch and fast-twitch oxidative fibers are capable of utilizing aerobic metabolism
FATIGUE
When we think of skeletal muscles getting tired, we often use the word fatigue, however, the physiological causes of fatigue vary considerably. At the simplest level, fatigue is used to describe a condition in which the muscle is no longer able to contract optimally. To make discussion easier, we will divide fatigue into two broad categories: Central fatigue and peripheral fatigue. Central fatigue describes the uncomfortable feelings that come from being tired, it is often called “psychological fatigue.” It has been suggested that central fatigue arises from factors released by the muscle during exercise that signal the brain to “feel” tired. Psychological fatigue precedes peripheral fatigue and occurs well before the muscle fiber can no longer contract. One of the outcomes of training is to learn how to overcome psychological fatigue. As we train we learn that those feelings are not so bad and that we can continue to perform even when it feels uncomfortable. For this reason, elite athletes hire trainers that push them and force them to move past the psychological fatigue.
Peripheral fatigue can occur anywhere between the neuromuscular junction and the contractile elements of the muscle. It can be divided into two subcategories, low frequency (marathon running) and high frequency (circuit training) fatigue. High frequency fatigue results from impaired membrane excitability as a result of imbalances of ions. Potential causes are inadequate functioning of the Na^{+}/K^{+} pump, subsequent inactivation of Na^{+} channels and impairment of Ca^{2+} channels. Muscles can recover quickly, usually within 30 minutes or less, following high frequency fatigue. Low frequency fatigue is correlated with impaired Ca^{2+} release, probably due to excitation coupling contraction problems. It is much more difficult to recover from low frequency fatigue, taking from 24 hours to 72 hours.
In addition, there are many other potential fatigue contributors, these include: accumulation of inorganic phosphates, hydrogen ion accumulation and subsequent pH change, glycogen depletion, and imbalances in K^{+}. Please note that factors that are not on the list are ATP and lactic acid, both of which do not contribute to fatigue. The reality is we still don’t know exactly what causes fatigue and much research is currently devoted to this topic.
SKELETAL MUSCLE FIBER TYPES
Classically, skeletal muscle fibers can be categorized according to their speed of contraction and their resistance to fatigue. These classifications are in the process of being revised, but the basic types include:
- Slow twitch oxidative (type I) muscle fibers,
- Fast-twitch oxidative-glycolytic (Type IIA) muscle fibers, and
- Fast-twitch glycolytic (Type IIX) fibers.
Fast-twitch (type II) fibers develop tension two to three times faster than slow-twitch (type I) fibers. How fast a fiber can contract is related to how long it takes for completion of the cross-bridge cycle. This variability is due to different varieties of myosin molecules and how quickly they can hydrolyze ATP. Recall that it is the myosin head that splits ATP. Fast-twitch fibers have a more rapid ATPase (splitting of ATP into ADP + P_{i}) ability. Fast-twitch fibers also pump Ca^{2+} ions back into the sarcoplasmic reticulum very quickly, so these cells have much faster twitches than the slower variety. Thus, fast-twitch fibers can complete multiple contractions much more rapidly than slow-twitch fibers. For a complete list of how muscle fibers differ in their ability to resist fatigue see the table below:
Slow Twitch Oxidative (Type I) | Fast-twitch Oxidative (Type IIA) | Fast-Twitch Glycolytic (Type IIX) | |
Myosin ATPase activity | slow | fast | fast |
Size (diameter) | small | medium | large |
Duration of contraction | long | short | short |
SERCA pump activity | slow | fast | fast |
Fatigue | resistant | resistant | easily fatigued |
Energy utilization | aerobic/oxidative | both | anerobic/glycolytic |
capillary density | high | medium | low |
mitochondria | high numbers | medium numbers | low numbers |
Color | red (contain myoglobin) | red (contain myoglobin) | white (no myoglobin) |
In human skeletal muscles, the ratio of the various fiber types differs from muscle to muscle. For example the gastrocnemius muscle of the calf contains about half slow and half fast type fibers, while the deeper calf muscle, the soleus, is predominantly slow twitch. On the other hand the eye muscles are predominantly fast twitch. As a result, the gastrocnemius muscle is used in sprinting while the soleus muscle is important for standing. In addition, women seem to have a higher ratio of slow twitch to fast twitch compared to men. The “preferred” fiber type for sprinting athletes is the fast-twitch glycolytic, which is very fast, however, most humans have a very low percentage of these fibers, < 1%. Muscle biopsies of one world class sprinter revealed 72% fast twitch fibers and amazingly 20% were type IIX. The Holy Grail of muscle research is to determine how to change skeletal muscle fibers from one type to another. It appears that muscle fiber types are determined embryologically by the type of neuron that innervates the muscle fiber. The default muscle appears to be slow, type I fibers. If a muscle is innervated by a small neuron that muscle fiber will remain slow, whereas large mylenated fibers induce the fast isoforms. In addition, the frequency of firing rates of the neuron also alters the muscle fiber type. Research suggests that humans have subtypes of fibers, making up about <5% of the muscle, that are dually innervated and allow for switching between slow and fast to occur. Generally, it would appear that genetics determine the type of innervation that occurs and subsequent muscle fiber types and that training may be able to slightly alter the ratios due to the dually innervated muscles. However, since <5% have dual innervation, genetics is going to play a much greater role in your fiber types than your training.
**You may use the buttons below to go to the next or previous reading in this Module**
Print this page
3 Tips for Building Fast Twitch Muscles
Sprint faster. Be more explosive off the line. Increase agility and quickness. These common goals cannot be achieved without the development of fast-twitch muscles. What are fast-twitch muscles, you ask? The difference between fast-twitch and slow-twitch muscles has a lot to do with the intensity of movement sustained over time.
If a person were to hold an abdominal plank for a two minutes, for example, or do a wall sit for five minutes, they would be developing and using (mostly) slow-twitch muscles. When it comes to slow-twitch, think endurance.
Fast-twitch muscle fibers, on the other hand, are activated by high intensity movements sustained in short bursts. Examples include sprints, burpees, and quick lateral movements. Many activities, such as boxing and basketball, incorporate both slow-twitch and fast-twitch fibers.
How to Build Fast-Twitch Muscle
In many ways, building fast-twitch muscles is about diversifying your workouts. The idea is to introduce activities that force the body to recruit fast-twitch muscle fibers it might not otherwise use. Here are three tips to help you do just that:
- Expand your strength training – Resistance training is an important cornerstone of most fitness regimens. Incorporate more fast-twitch movements by performing reps at a faster rate, or working in exercises like power clean and snatch.
- Sprints and agility drills – Straight sprints can be quite boring. Try adding changes in motion to your sprint routine, such as there-backs or three-point agility drills. Sprint up and down a flight of stairs. Incorporate resistance bands or perform explosive movements underwater. You can also recruit new muscle fibers by borrowing from sports you don’t even play, but that rely on good agility, such as football, soccer, and gymnastics.
- Work in some plyometrics – Plyometrics are all about quick, powerful expansions and contractions of a given muscle or muscle group. The burpee is a classic (and timeless!) example. You might also consider explosive bodyweight exercises such as jump squats, split-squat lunges, or plyo push-up. Military training and crossfit programs are famous for incorporating plyometric exercises, so start there if you need ideas.
Pushing the Boundaries Safely
You’ll notice that building fast-twitch muscle fibers often requires pushing your body beyond the limits you are used to. While beneficial in many ways, this also introduces increased injury risk.
Athletes of the highest levels often perform these exercises in controlled environments, or under the supervision of certified professionals. Tools like the AlterG® Anti-Gravity Treadmill™ can be used, for example, to limit injury risk and body-weight impact during highly strenuous sprinting exercises.
Regardless of how you choose to build fast-twitch muscles, remember that no workout regimen is exclusive of your diet and sleep regimen. Your ability to perform these workouts without injury, as well as to recover properly after taxing workouts, is just as important to building fast-twitch muscle fibers as the exercises themselves.
Abbreviated multiplication formulas 💣
Abbreviated multiplication formulas
Instead of letters a, b, there can be any numbers, variables or even integer expressions. To quickly solve problems, it is better to learn the basic 7 formulas of abbreviated multiplication (FSO) by heart. Yes, algebra is like that, you need to be ready to memorize a lot.
Below is a handy plate that can be printed and used as a bookmark for quickly memorizing formulas.
How to read abbreviated multiplication formulas
Learning to pronounce the formulas of the abbreviated expression:
- Difference of squares of two expressions is equal to the product of their difference and their sum.
- The square of the sum of of two expressions is equal to the square of the first plus twice the product of the first by the second plus the square of the second.
- The square of the difference of the two expressions is equal to the square of the first minus twice the product of the first by the second plus the square of the second.
- The sum of the cubes of two expressions is equal to the product of the sum of the first and the second by the incomplete square of their difference.
- The difference between the cubes of the two expressions is equal to the product of the difference between the first and the second by the incomplete square of their sum.
- The cube of the sum of of two expressions is equal to the cube of the first plus three times the square of the first plus three times the square of the second plus the cube of the second.
- The cube of the difference of the two expressions is equal to the cube of the first minus the triple product of the square of the first by the second plus triple the product of the first and the square of the second minus the cube of the second.
Proof of reduced multiplication formulas
Recall that the difference between the squares of two numbers a and b is equal to the product of their difference and their sum: a ^{ 2 } – b ^{ 2 } = (a – b) * (a + b).
In other words, the product of the sum of a and b by their difference is equal to the difference of their squares: (a – b) * (a + b) = a ^{ 2 } – b ^{ 2 }.
It is important to know that the difference of squares is not equal to the square of the difference: a ^{ 2 } – b ^{ 2 } ≠ (a – b) ^{ 2 }.
Prove that a ^{ 2 } – b ^{ 2 } = (a – b) * (a + b).
Let’s go:
- Using the artificial method, add and subtract the same a * b.
+ a * b – a * b = 0
a ^{ 2 } – b ^{ 2 } = a ^{ 2 } – b ^{ 2 } + ab – ab
- Let’s group it differently: a ^{ 2 } – b ^{ 2 } + a * b – a * b = a ^{ 2 } – a * b + a * b – b ^{ 2 }
- Continue to group: a ^{ 2 } – a * b – b ^{ 2 } + a * b = (a ^{ 2 } – a * b) + (a * b – b ^{ 2 })
- Pull out the common factors outside the parentheses:
(a ^{ 2 } – a * b) + (a * b – b ^{ 2 }) = a * (a – b) + b * (a – b)
- Factor out (a – b).a * (a – b) + b * (a – b) = (a – b) * (a + b)
- Result of proof: a ^{ 2 } – b ^{ 2 } = (a – b) * (a + b)
- In order to prove in the opposite direction: (a – b) * (a + b) = a ^{ 2 } – b ^{ 2 }, you need to expand the parentheses: (a – b) * (a + b) = a * a + a * b – b * a – b * b = a ^{ 2 } – b ^{ 2 }.
The rest of the FSO can be proved by a similar method.
Additional abbreviated multiplication formulas
A few more important identities should be added to the table of the main FSOs, which will be useful for solving problems.
Binomial Newton
Formula for the decomposition into separate terms of a non-negative integer power of the sum of two variables. It is written like this:
An example of calculating binomial coefficients that stand in line n in Pascal’s triangle:
FSU for the square and the cube of the sum and difference are special cases of the Newton binomial formula for n = 2 and n = 3.
Formula for squaring the sum of three, four or more terms
It will come in handy if there are more than two terms in the sum to be raised to a power.
(a _{ 1 } + a _{ 2 } +… + a _{ n }) ^{ 2 } = a _{ 1 } ^{ 2 } + a _{ 2 } ^{ 2 } +… + a _{ n-1 } ^{ 2 } + a _{ n } ^{ 2 } + 2 * a _{ 1 } * a _{ 2 } + 2 * a _{ 1 } * a _{ 3 } + 2 * a _{ 1 } * a _{ 4 } +… +
+ 2 * a _{ 1 } * a _{ n-1 } + 2 * a _{ 1 } * a _{ n } + 2 * a _{ 2 } * a _{ 3 } + 2 * a _{ 2 } * a _{ 4 } +… + 2 * a _{ 2 } * a _{ n-1 } + 2 * a _{ 2 } * a _{ n } +… +
+ 2 * a _{ n-1 } * a _{ n }
It reads like this: the square of the sum of n terms is equal to the sum of the squares of all these terms and the doubled products of all possible pairs of these terms.
Formula for the difference between the n-th powers of two terms
a ^{ n } – b ^{ n } = (a – b) * (a ^{ n-1 } + a ^{ n-2 } * b + a ^{ n-3 } * b ^{ 2 } +… + a * b ^{ n-2 } + b ^{ n-1 }).
For even indicators, you can write it like this:
a ^{ 2 * m } – b ^{ 2 * m } = (a ^{ 2 } – b ^{ 2 }) * (a ^{ 2 * m − 2 } + a ^{ 2 * m − 4 } * b ^{ 2 } + a ^{ 2 * m − 6 } * b ^{ 4 } +… + b ^{ 2 * m − 2 }).
For odd exponents:
a ^{ 2 * m + 1 } – b ^{ 2 * m + 1 } = (a – b) * (a ^{ 2 * m } + a ^{ 2 * m − 1 } * b + a ^{ 2 * m −2 } * b ^{ 2 } +… + b ^{ 2 * m }).
Special cases are formulas for the difference between squares and cubes for n = 2 and n = 3. For the difference between cubes, b can also be replaced by −b.
Problem solving
Let’s practice and see examples with fractions.
Task 1
What to do: Calculate the square of the product (55 + 10) ^{ 2 }.
How to solve: we use the formula for the square of the sum: (55 + 10) ^{ 2 } = 55 ^{ 2 } + 2 * 55 * 10 + 10 ^{ 2 } = 3025 + 1100 + 100 = 4225.
Task 2
How to: Simplify 64 * s ^{ 3 } – 8.
How to solve: apply the difference of cubes: 64 * s ^{ 3 } – 8 = (4 * s) ^{ 3 } – 2 ^{ 3 } = (4 * s – 2) ((4 * s) ^{ 2 } + 4 * s * 2 + 2 ^{ 2 }) = (4 * s – 2) (16 * s ^{ 2 } + 8 * s + 4).
Task 3
What to do: Expand parentheses (7 * y – x) * (7 * y + x).
How we solve:
- Let’s multiply: (7 * y – x) * (7 * y + x) = 7 * y * 7 * y + 7 * y * x – x * 7 * y – x * x = 49 * y ^{ 2 } + 7 * y * x – 7 * y * x – x ^{ 2 } = 49 * y ^{ 2 } – x ^{ 2 }.
- Use the abbreviated multiplication formula: (7 * y – x) * (7 * y + x) = (7 * y) ^{ 2 } – x ^{ 2 } = 49 * y ^{ 2 } – x ^{ 2 }.
You shouldn’t be afraid of polynomials, just do every action in sequence. Solving problems with formulas is faster and more convenient – keep the cheat sheet, remember and please your teachers 🙂
Abbreviated multiplication formulas: table, examples of use
Abbreviated Multiplication Formulas (FSF) are used for exponentiation and multiplication of numbers and expressions.Often these formulas allow computations to be performed more compactly and quickly.
In this article, we will list the basic formulas for abbreviated multiplication, group them in a table, consider examples of using these formulas, and also dwell on the principles of proof of abbreviated multiplication formulas.
Abbreviated multiplication formulas. Table
For the first time, the topic of FSU is considered within the framework of the course “Algebra” for the 7th grade. Below are 7 basic formulas.
Abbreviated multiplication formulas
- Sum square formula: a + b2 = a2 + 2ab + b2
- Difference square formula: a-b2 = a2-2ab + b2
- Sum cube formula: a + b3 = a3 + 3a2b + 3ab2 + b3
- Difference cube formula: a-b3 = a3-3a2b + 3ab2-b3
- Difference of squares formula: a2-b2 = a-ba + b
- the formula for the sum of cubes: a3 + b3 = a + ba2-ab + b2
- Cubes difference formula: a3-b3 = a-ba2 + ab + b2
The letters a, b, c in these expressions can be any numbers, variables or expressions.For ease of use, it is best to learn the seven basic formulas by heart. Let’s summarize them in a table and give them below, encircling them with a frame.
The first four formulas allow you to calculate, respectively, the square or cube of the sum or difference of two expressions.
The fifth formula calculates the difference of squares of expressions by the product of their sum and difference.
The sixth and seventh formulas are, respectively, the multiplication of the sum and the difference of the expressions by the incomplete square of the difference and the incomplete square of the sum.
The formula of abbreviated multiplication is sometimes also called the identities of abbreviated multiplication. This is not surprising, since every equality is an identity.
When solving practical examples, abbreviated multiplication formulas with rearranged left and right sides are often used. This is especially convenient when a factorization of a polynomial takes place.
Additional abbreviated multiplication formulas
We will not limit ourselves to the 7th grade course in algebra and add a few more formulas to our FSU table.
First, consider the Newton binomial formula.
a + bn = Cn0 an + Cn1 an-1 b + Cn2 an-2 b2 + .. + Cnn-1 a bn-1 + Cnn bn
Here Cnk are binomial coefficients that appear in line n in the pascal triangle. Binomial coefficients are calculated using the formula:
Cnk = n! K! (N-k)! = N (n-1) (n-2) .. (n- (k-1)) k!
As you can see, the FSE for the square and the cube of the difference and the sum is a special case of the Newton binomial formula for n = 2 and n = 3, respectively.
But what if there are more than two terms in the sum to be raised to the power? The formula for the square of the sum of three, four or more terms will be useful.
a1 + a2 + .. + an2 = a12 + a22 + .. + an2 + 2a1a2 + 2a1a3 + .. + 2a1an + 2a2a3 + 2a2a4 + .. + 2a2an + 2an-1an
How to read this formula? The square of the sum of n terms is equal to the sum of the squares of all terms and doubled products of all possible pairs of these terms.
Another formula that may come in handy is the formula for the difference between the n-th powers of two terms.
an-bn = a-ban-1 + an-2b + an-3b2 + .. + a2bn-2 + bn-1
This formula is usually divided into two formulas – respectively for even and odd degrees.
For even numbers 2m:
a2m-b2m = a2-b2a2m-2 + a2m-4b2 + a2m-6b4 + .. + b2m-2
For odd exponents 2m + 1:
a2m + 1-b2m + 1 = a2-b2a2m + a2m-1b + a2m-2b2 + .. + b2m
The formulas for the difference of squares and the difference between cubes, you guessed it, are special cases of this formula for n = 2 and n = 3, respectively. For the difference of cubes, b is also replaced with -b.
How to read abbreviated multiplication formulas?
Let’s give the appropriate wording for each formula, but first let’s figure out the principle of reading formulas.The most convenient way to do this is by example. Let’s take the very first formula for the square of the sum of two numbers.
a + b2 = a2 + 2ab + b2.
They say: the square of the sum of two expressions a and b is equal to the sum of the square of the first expression, twice the product of the expressions, and the square of the second expression.
All other formulas are read in the same way. For the square of the difference a-b2 = a2-2ab + b2 we write:
the square of the difference between the two expressions a and b is equal to the sum of the squares of these expressions minus twice the product of the first and second expressions.
Do you need a teacher’s help?
Describe the task – and our experts will help you!
Describe the task
Let’s read the formula a + b3 = a3 + 3a2b + 3ab2 + b3. The cube of the sum of two expressions a and b is equal to the sum of the cubes of these expressions, three times the square of the first expression by the second, and three times the square of the second expression by the first expression.
We proceed to reading the formula for the difference between the cubes a-b3 = a3-3a2b + 3ab2-b3. The cube of the difference of two expressions a and b is equal to the cube of the first expression minus three times the square of the first expression and the second, plus three times the square of the second expression and the first expression, minus the cube of the second expression.
The fifth formula a2-b2 = a-ba + b (difference of squares) reads as follows: the difference of the squares of two expressions is equal to the product of the difference and the sum of the two expressions.
Expressions like a2 + ab + b2 and a2-ab + b2 for convenience are called, respectively, the incomplete square of the sum and the incomplete square of the difference.
With this in mind, the formulas for the sum and difference of the cubes read as follows:
The sum of the cubes of two expressions is equal to the product of the sum of these expressions by the incomplete square of their difference.
The difference between the cubes of two expressions is equal to the product of the difference between these expressions and the incomplete square of their sum.
Proof of FSO
It is quite easy to prove the FSO. Based on the properties of multiplication, we multiply the parts of the formulas in parentheses.
As an example, consider the formula for the square of the difference.
a-b2 = a2-2ab + b2.
To raise an expression to the second power, you need to multiply this expression by itself.
a-b2 = a-ba-b.
Open brackets:
a-ba-b = a2-ab-ba + b2 = a2-2ab + b2.
The formula is proven. The rest of the FSOs are proved in a similar way.
Examples of FSU application
The purpose of using abbreviated multiplication formulas is to multiply and exponentiate quickly and quickly. However, this is not the entire scope of the FSO. They are widely used in abbreviating expressions, reducing fractions, factoring polynomials. Here are some examples.
Example 1. FSO
Simplify the expression 9y- (1 + 3y) 2.
Apply the formula for the sum of squares and get:
9y- (1 + 3y) 2 = 9y- (1 + 6y + 9y2) = 9y-1-6y-9y2 = 3y-1-9y2
Example 2.FSU
Reduce the fraction 8×3-z64x2-z4.
Note that the expression in the numerator is the difference between the cubes, and the denominator is the difference in the squares.
8×3-z64x2-z4 = 2x-z (4×2 + 2xz + z4) 2x-z2x + z.
Reduce and get:
8×3-z64x2-z4 = (4×2 + 2xz + z4) 2x + z
Also FSO help to calculate the values of expressions. The main thing is to be able to notice where to apply the formula. Let’s show this with an example.
Let’s square the number 79. Instead of cumbersome calculations, we will write:
79 = 80-1; 792 = 80-12 = 6400-160 + 1 = 6241.
It would seem that a complex calculation was carried out quickly with just the use of the abbreviated multiplication formulas and the multiplication table.
Another important point is the selection of the square of the binomial. The expression 4×2 + 4x-3 can be converted to 2×2 + 2 · 2 · x · 1 + 12-4 = 2x + 12-4. Such transformations are widely used in integration.
Abbreviated multiplication formulas: table, examples of use
Abbreviated Multiplication Formulas (FSF) are used for exponentiation and multiplication of numbers and expressions.Often these formulas allow computations to be performed more compactly and quickly.
In this article, we will list the basic formulas for abbreviated multiplication, group them in a table, consider examples of using these formulas, and also dwell on the principles of proof of abbreviated multiplication formulas.
Abbreviated multiplication formulas. Table
For the first time, the topic of FSU is considered within the framework of the course “Algebra” for the 7th grade. Below are 7 basic formulas.
Abbreviated multiplication formulas
- Sum square formula: a + b2 = a2 + 2ab + b2
- Difference square formula: a-b2 = a2-2ab + b2
- Sum cube formula: a + b3 = a3 + 3a2b + 3ab2 + b3
- Difference cube formula: a-b3 = a3-3a2b + 3ab2-b3
- Difference of squares formula: a2-b2 = a-ba + b
- the formula for the sum of cubes: a3 + b3 = a + ba2-ab + b2
- Cubes difference formula: a3-b3 = a-ba2 + ab + b2
The letters a, b, c in these expressions can be any numbers, variables or expressions.For ease of use, it is best to learn the seven basic formulas by heart. Let’s summarize them in a table and give them below, encircling them with a frame.
The first four formulas allow you to calculate, respectively, the square or cube of the sum or difference of two expressions.
The fifth formula calculates the difference of squares of expressions by the product of their sum and difference.
The sixth and seventh formulas are, respectively, the multiplication of the sum and the difference of the expressions by the incomplete square of the difference and the incomplete square of the sum.
The formula of abbreviated multiplication is sometimes also called the identities of abbreviated multiplication. This is not surprising, since every equality is an identity.
When solving practical examples, abbreviated multiplication formulas with rearranged left and right sides are often used. This is especially convenient when a factorization of a polynomial takes place.
Additional abbreviated multiplication formulas
We will not limit ourselves to the 7th grade course in algebra and add a few more formulas to our FSU table.
First, consider the Newton binomial formula.
a + bn = Cn0 an + Cn1 an-1 b + Cn2 an-2 b2 + .. + Cnn-1 a bn-1 + Cnn bn
Here Cnk are binomial coefficients that appear in line n in the pascal triangle. Binomial coefficients are calculated using the formula:
Cnk = n! K! (N-k)! = N (n-1) (n-2) .. (n- (k-1)) k!
As you can see, the FSE for the square and the cube of the difference and the sum is a special case of the Newton binomial formula for n = 2 and n = 3, respectively.
But what if there are more than two terms in the sum to be raised to the power? The formula for the square of the sum of three, four or more terms will be useful.
a1 + a2 + .. + an2 = a12 + a22 + .. + an2 + 2a1a2 + 2a1a3 + .. + 2a1an + 2a2a3 + 2a2a4 + .. + 2a2an + 2an-1an
How to read this formula? The square of the sum of n terms is equal to the sum of the squares of all terms and doubled products of all possible pairs of these terms.
Another formula that may come in handy is the formula for the difference between the n-th powers of two terms.
an-bn = a-ban-1 + an-2b + an-3b2 + .. + a2bn-2 + bn-1
This formula is usually divided into two formulas – respectively for even and odd degrees.
For even numbers 2m:
a2m-b2m = a2-b2a2m-2 + a2m-4b2 + a2m-6b4 + .. + b2m-2
For odd exponents 2m + 1:
a2m + 1-b2m + 1 = a2-b2a2m + a2m-1b + a2m-2b2 + .. + b2m
The formulas for the difference of squares and the difference between cubes, you guessed it, are special cases of this formula for n = 2 and n = 3, respectively. For the difference of cubes, b is also replaced with -b.
How to read abbreviated multiplication formulas?
Let’s give the appropriate wording for each formula, but first let’s figure out the principle of reading formulas.The most convenient way to do this is by example. Let’s take the very first formula for the square of the sum of two numbers.
a + b2 = a2 + 2ab + b2.
They say: the square of the sum of two expressions a and b is equal to the sum of the square of the first expression, twice the product of the expressions, and the square of the second expression.
All other formulas are read in the same way. For the square of the difference a-b2 = a2-2ab + b2 we write:
the square of the difference between the two expressions a and b is equal to the sum of the squares of these expressions minus twice the product of the first and second expressions.
Do you need a teacher’s help?
Describe the task – and our experts will help you!
Describe the task
Let’s read the formula a + b3 = a3 + 3a2b + 3ab2 + b3. The cube of the sum of two expressions a and b is equal to the sum of the cubes of these expressions, three times the square of the first expression by the second, and three times the square of the second expression by the first expression.
We proceed to reading the formula for the difference between the cubes a-b3 = a3-3a2b + 3ab2-b3. The cube of the difference of two expressions a and b is equal to the cube of the first expression minus three times the square of the first expression and the second, plus three times the square of the second expression and the first expression, minus the cube of the second expression.
The fifth formula a2-b2 = a-ba + b (difference of squares) reads as follows: the difference of the squares of two expressions is equal to the product of the difference and the sum of the two expressions.
Expressions like a2 + ab + b2 and a2-ab + b2 for convenience are called, respectively, the incomplete square of the sum and the incomplete square of the difference.
With this in mind, the formulas for the sum and difference of the cubes read as follows:
The sum of the cubes of two expressions is equal to the product of the sum of these expressions by the incomplete square of their difference.
The difference between the cubes of two expressions is equal to the product of the difference between these expressions and the incomplete square of their sum.
Proof of FSO
It is quite easy to prove the FSO. Based on the properties of multiplication, we multiply the parts of the formulas in parentheses.
As an example, consider the formula for the square of the difference.
a-b2 = a2-2ab + b2.
To raise an expression to the second power, you need to multiply this expression by itself.
a-b2 = a-ba-b.
Open brackets:
a-ba-b = a2-ab-ba + b2 = a2-2ab + b2.
The formula is proven. The rest of the FSOs are proved in a similar way.
Examples of FSU application
The purpose of using abbreviated multiplication formulas is to multiply and exponentiate quickly and quickly. However, this is not the entire scope of the FSO. They are widely used in abbreviating expressions, reducing fractions, factoring polynomials. Here are some examples.
Example 1. FSO
Simplify the expression 9y- (1 + 3y) 2.
Apply the formula for the sum of squares and get:
9y- (1 + 3y) 2 = 9y- (1 + 6y + 9y2) = 9y-1-6y-9y2 = 3y-1-9y2
Example 2.FSU
Reduce the fraction 8×3-z64x2-z4.
Note that the expression in the numerator is the difference between the cubes, and the denominator is the difference in the squares.
8×3-z64x2-z4 = 2x-z (4×2 + 2xz + z4) 2x-z2x + z.
Reduce and get:
8×3-z64x2-z4 = (4×2 + 2xz + z4) 2x + z
Also FSO help to calculate the values of expressions. The main thing is to be able to notice where to apply the formula. Let’s show this with an example.
Let’s square the number 79. Instead of cumbersome calculations, we will write:
79 = 80-1; 792 = 80-12 = 6400-160 + 1 = 6241.
It would seem that a complex calculation was carried out quickly with just the use of the abbreviated multiplication formulas and the multiplication table.
Another important point is the selection of the square of the binomial. The expression 4×2 + 4x-3 can be converted to 2×2 + 2 · 2 · x · 1 + 12-4 = 2x + 12-4. Such transformations are widely used in integration.
Abbreviated multiplication formulas: table, examples of use
Abbreviated Multiplication Formulas (FSF) are used for exponentiation and multiplication of numbers and expressions.Often these formulas allow computations to be performed more compactly and quickly.
In this article, we will list the basic formulas for abbreviated multiplication, group them in a table, consider examples of using these formulas, and also dwell on the principles of proof of abbreviated multiplication formulas.
Abbreviated multiplication formulas. Table
For the first time, the topic of FSU is considered within the framework of the course “Algebra” for the 7th grade. Below are 7 basic formulas.
Abbreviated multiplication formulas
- Sum square formula: a + b2 = a2 + 2ab + b2
- Difference square formula: a-b2 = a2-2ab + b2
- Sum cube formula: a + b3 = a3 + 3a2b + 3ab2 + b3
- Difference cube formula: a-b3 = a3-3a2b + 3ab2-b3
- Difference of squares formula: a2-b2 = a-ba + b
- the formula for the sum of cubes: a3 + b3 = a + ba2-ab + b2
- Cubes difference formula: a3-b3 = a-ba2 + ab + b2
The letters a, b, c in these expressions can be any numbers, variables or expressions.For ease of use, it is best to learn the seven basic formulas by heart. Let’s summarize them in a table and give them below, encircling them with a frame.
The first four formulas allow you to calculate, respectively, the square or cube of the sum or difference of two expressions.
The fifth formula calculates the difference of squares of expressions by the product of their sum and difference.
The sixth and seventh formulas are, respectively, the multiplication of the sum and the difference of the expressions by the incomplete square of the difference and the incomplete square of the sum.
The formula of abbreviated multiplication is sometimes also called the identities of abbreviated multiplication. This is not surprising, since every equality is an identity.
When solving practical examples, abbreviated multiplication formulas with rearranged left and right sides are often used. This is especially convenient when a factorization of a polynomial takes place.
Additional abbreviated multiplication formulas
We will not limit ourselves to the 7th grade course in algebra and add a few more formulas to our FSU table.
First, consider the Newton binomial formula.
a + bn = Cn0 an + Cn1 an-1 b + Cn2 an-2 b2 + .. + Cnn-1 a bn-1 + Cnn bn
Here Cnk are binomial coefficients that appear in line n in the pascal triangle. Binomial coefficients are calculated using the formula:
Cnk = n! K! (N-k)! = N (n-1) (n-2) .. (n- (k-1)) k!
As you can see, the FSE for the square and the cube of the difference and the sum is a special case of the Newton binomial formula for n = 2 and n = 3, respectively.
But what if there are more than two terms in the sum to be raised to the power? The formula for the square of the sum of three, four or more terms will be useful.
a1 + a2 + .. + an2 = a12 + a22 + .. + an2 + 2a1a2 + 2a1a3 + .. + 2a1an + 2a2a3 + 2a2a4 + .. + 2a2an + 2an-1an
How to read this formula? The square of the sum of n terms is equal to the sum of the squares of all terms and doubled products of all possible pairs of these terms.
Another formula that may come in handy is the formula for the difference between the n-th powers of two terms.
an-bn = a-ban-1 + an-2b + an-3b2 + .. + a2bn-2 + bn-1
This formula is usually divided into two formulas – respectively for even and odd degrees.
For even numbers 2m:
a2m-b2m = a2-b2a2m-2 + a2m-4b2 + a2m-6b4 + .. + b2m-2
For odd exponents 2m + 1:
a2m + 1-b2m + 1 = a2-b2a2m + a2m-1b + a2m-2b2 + .. + b2m
The formulas for the difference of squares and the difference between cubes, you guessed it, are special cases of this formula for n = 2 and n = 3, respectively. For the difference of cubes, b is also replaced with -b.
How to read abbreviated multiplication formulas?
Let’s give the appropriate wording for each formula, but first let’s figure out the principle of reading formulas.The most convenient way to do this is by example. Let’s take the very first formula for the square of the sum of two numbers.
a + b2 = a2 + 2ab + b2.
They say: the square of the sum of two expressions a and b is equal to the sum of the square of the first expression, twice the product of the expressions, and the square of the second expression.
All other formulas are read in the same way. For the square of the difference a-b2 = a2-2ab + b2 we write:
the square of the difference between the two expressions a and b is equal to the sum of the squares of these expressions minus twice the product of the first and second expressions.
Do you need a teacher’s help?
Describe the task – and our experts will help you!
Describe the task
Let’s read the formula a + b3 = a3 + 3a2b + 3ab2 + b3. The cube of the sum of two expressions a and b is equal to the sum of the cubes of these expressions, three times the square of the first expression by the second, and three times the square of the second expression by the first expression.
We proceed to reading the formula for the difference between the cubes a-b3 = a3-3a2b + 3ab2-b3. The cube of the difference of two expressions a and b is equal to the cube of the first expression minus three times the square of the first expression and the second, plus three times the square of the second expression and the first expression, minus the cube of the second expression.
The fifth formula a2-b2 = a-ba + b (difference of squares) reads as follows: the difference of the squares of two expressions is equal to the product of the difference and the sum of the two expressions.
Expressions like a2 + ab + b2 and a2-ab + b2 for convenience are called, respectively, the incomplete square of the sum and the incomplete square of the difference.
With this in mind, the formulas for the sum and difference of the cubes read as follows:
The sum of the cubes of two expressions is equal to the product of the sum of these expressions by the incomplete square of their difference.
The difference between the cubes of two expressions is equal to the product of the difference between these expressions and the incomplete square of their sum.
Proof of FSO
It is quite easy to prove the FSO. Based on the properties of multiplication, we multiply the parts of the formulas in parentheses.
As an example, consider the formula for the square of the difference.
a-b2 = a2-2ab + b2.
To raise an expression to the second power, you need to multiply this expression by itself.
a-b2 = a-ba-b.
Open brackets:
a-ba-b = a2-ab-ba + b2 = a2-2ab + b2.
The formula is proven. The rest of the FSOs are proved in a similar way.
Examples of FSU application
The purpose of using abbreviated multiplication formulas is to multiply and exponentiate quickly and quickly. However, this is not the entire scope of the FSO. They are widely used in abbreviating expressions, reducing fractions, factoring polynomials. Here are some examples.
Example 1. FSO
Simplify the expression 9y- (1 + 3y) 2.
Apply the formula for the sum of squares and get:
9y- (1 + 3y) 2 = 9y- (1 + 6y + 9y2) = 9y-1-6y-9y2 = 3y-1-9y2
Example 2.FSU
Reduce the fraction 8×3-z64x2-z4.
Note that the expression in the numerator is the difference between the cubes, and the denominator is the difference in the squares.
8×3-z64x2-z4 = 2x-z (4×2 + 2xz + z4) 2x-z2x + z.
Reduce and get:
8×3-z64x2-z4 = (4×2 + 2xz + z4) 2x + z
Also FSO help to calculate the values of expressions. The main thing is to be able to notice where to apply the formula. Let’s show this with an example.
Let’s square the number 79. Instead of cumbersome calculations, we will write:
79 = 80-1; 792 = 80-12 = 6400-160 + 1 = 6241.
It would seem that a complex calculation was carried out quickly with just the use of the abbreviated multiplication formulas and the multiplication table.
Another important point is the selection of the square of the binomial. The expression 4×2 + 4x-3 can be converted to 2×2 + 2 · 2 · x · 1 + 12-4 = 2x + 12-4. Such transformations are widely used in integration.
Abbreviated multiplication formulas
Abbreviated multiplication formulas
Abbreviated multiplication formulas are used in mathematics, or rather in algebra, to quickly obtain the result of some algebraic expressions. formulas for reduced multiplication are obtained from the algebraic rules for multiplying polynomials. The use of formulas for reduced multiplication allows you to more quickly solve mathematical problems, reduce cumbersome algebraic expressions. The rules of algebra allow you to arbitrarily perform transformations of expressions using abbreviated multiplication formulas: you can represent the left side of the equality as the right side or transform the right side of the equality as the left side of the equality.It is recommended to know the abbreviated multiplication formulas by heart, since they are often used in solving problems and equations in algebra and mathematics. The most common formulas are the first three abbreviated multiplication formulas .
It is recommended to save the above figure on your computer as a cheat sheet for mathematics and algebra. The formulas shown in the figure are not a complete list of abbreviated multiplication formulas. In algebra, there are other formulas for abbreviated multiplication and division.All of these formulas have their own names. Let us consider in more detail the names of the above formulas for abbreviated multiplication.
The first [1] in the picture is the square of the sum. The square of the sum of is equal to the square of the first term of the binomial plus twice the product of the first term by the second term of the binomial plus the square of the second term of the binomial:
(a + b) ² = a² + 2ab + b²
The second [2] The formula for abbreviated multiplication is called the square of the difference. The square of the difference is the square of the first term minus twice the product of the first term and the second term of the binomial plus the square of the second term of the binomial. This formula is very similar to the formula for the square of the sum and differs only in the sign before the doubled product:
(a – b) ² = a² – 2ab + b²
In general terms, the square of the sum and the square of the difference can be written as follows:
(a ± b) ² = a² ± 2ab + b²
Formula number three [3] is called the difference of squares. Difference of squares is equal to the sum of the first two terms of the binomial multiplied by the difference of the first and second terms of the binomial:
a² – b² = (a + b) · (a – b)
The fourth formula [4] is called the cube of the sum. The cube of the sum is equal to the sum of the cubes of the first and second terms of the binomial, triple products of the square of the first term of the binomial by the second, and the square of the second term of the binomial by the first:
(a + b) ³ = a³ + 3a²b + 3b²a + b³
Fifth [5] The formula is similar to the sum cube and is called the difference cube. The cube of the difference is equal to the cube of the first term of the binomial minus three times the square of the first term of the binomial plus three times the product of the first term of the binomial by the square of the second minus the cube of the second term of the binomial:
(a – b) ³ = a³ – 3a²b + 3b²a – b³
With one formula, the cube of the sum and the cube of the difference can be written using the plus-minus signs:
(a ± b) ³ = a³ ± 3a²b + 3b²a ± b³
Sixth [6] The formula is called the sum of cubes. The sum of cubes is equal to the sum of the first and second terms of the binomial multiplied by the square of the first term of the binomial minus the product of the first and second terms of the binomial plus the square of the second term of the binomial:
a³ + b³ = (a + b) · (a² – 2ab + b²)
Seventh [7] the formula is similar to the previous one and is called the difference of cubes. The difference of cubes is equal to the difference of the first and second terms of the binomial multiplied by the square of the first term of the binomial plus the product of the first and second terms of the binomial plus the square of the second term of the binomial:
a³ – b³ = (a – b) · (a² + 2ab + b²)
In one formula, the sum cube and the difference cube can be written using the plus-minus and minus-plus signs.
If you liked the publication and would like to know more, please help me with other materials.
August 9, 2010 – September 22, 2019 .
© 2006 – 2021 Nikolay Khizhnyak. All rights reserved.
Abbreviated multiplication formulas with examples
Abbreviated multiplication formulas (FSC) call several of the most common cases of multiplication of polynomials in practice.2 \)
Solution :
Notice how much faster and with less effort the result was obtained in the second case. And when you master this and other formulas to automatism, it will be even faster: you can just write the answer right away. Therefore, they are called ABBREVIATED multiplication formulas. So, knowing them and learning how to apply is definitely worth it.
Just in case, we note that any expressions can be used as \ (a \) and \ (b \) – the principle remains the same.2 + 4a = \)
Now we present similar terms.
\ (= – 8a + 9 = \)
Now we substitute and enjoy the simplicity of the calculations.
\ (= – 8 \ frac {17} {8} + 9 = -17 + 9 = 8 \)
We write the answer.2} {x-3} \) \ (= \) \ (\ frac {(x + 3) (x-3)} {x-3} \) \ (= \)
Now all the pros and cons are hidden in brackets, which means we can shorten the same brackets without any problems.
\ (= x + 3 \)
The answer is ready.
Answer: \ (x + 3 \).3) \)
The answer is ready.
These are the three basic formulas that you need to know be sure to ! There are also formulas with cubes (see above), it is also advisable to remember them or be able to quickly deduce them. We also note that in practice we often encounter several such formulas at once in one problem – this is normal. Just get in the habit of noticing formulas and applying them carefully and you will be fine.2} {x-2y + 3} \) \ (= \)
Once again, we carefully look at the numerator … think … think … and notice the formula for the difference of squares, in which \ (a = (x-2y) \), \ (b = 3 \). We lay out on it to the product of two brackets.
\ (\ frac {(x-2y-3) (x-2y + 3)} {x-2y + 3} \) \ (= \)
And now we are reducing the second parenthesis of the numerator and the entire denominator.
\ (x-2y-3 \)
The answer is ready.
Rapid multiplication formulas. Online calculator. Simplification of a polynomial. Multiplication of polynomials
Mathematical expressions (formulas) abbreviated multiplication
(the square of the sum and the difference, the cube of the sum and the difference, the difference of the squares, the sum and the difference of the cubes) are extremely irreplaceable in many areas of the exact sciences.These 7 symbolic notations are irreplaceable for simplifying expressions, solving equations, multiplying polynomials, canceling fractions, solving integrals and much more. This means that it will be very useful to understand how they are obtained, what they are for, and most importantly, how to remember them and then apply them. Then applying the abbreviated multiplication formula
in practice, the most difficult thing will be to see that there is x
and what do u have. Obviously no limit for a
and b
no, which means it can be any numeric or literal expressions.
And so they are:
First x 2
– at 2
= (x – y) (x + y)
. To calculate the difference of squares
two expressions must be multiplied by the differences of these expressions by their sums.
Second (x + y) 2
= x 2
+ 2xy + y 2
. To find squared sum
two expressions, you need to add the double product of the first expression to the second plus the square of the second expression to the square of the first expression.
Third (x – y) 2
= x 2
– 2xy + y 2
. To calculate squared difference
two expressions, you need to subtract the double product of the first expression by the second plus the square of the second expression from the square of the first expression.
Fourth (x + y) 3
= x 3
+ 3x 2 y + 3x 2
+ y 3.
To calculate cube sum
two expressions, you need to add to the cube of the first expression the triple product of the square of the first expression by the second plus triple the product of the first expression by the square of the second plus the cube of the second expression.
Fifth (x – y) 3
= x 3
– 3x 2 y + 3x 2
– at 3
. To calculate the cube difference
two expressions, it is necessary to subtract from the cube of the first expression the triple product of the square of the first expression by the second plus triple the product of the first expression by the square of the second minus the cube of the second expression.
Sixth x 3
+ y 3
= (x + y) (x 2
– xy + y 2)
To calculate the sum of cubes
two expressions, you need to multiply the sums of the first and second expressions by the incomplete square of the difference between these expressions.
Seventh x 3
– at 3
= (x – y) (x 2
+ xy + y 2)
To calculate Cube Difference
two expressions must be multiplied by the difference between the first and second expressions by the incomplete square of the sum of these expressions.
It is not difficult to remember that all formulas are applied to perform calculations and in the opposite direction (from right to left).
On the existence of these regularities
poured about 4 thousand years ago.They were widely used by the inhabitants of ancient Babylon and Egypt. But in those times they were expressed verbally or geometrically and did not use letters in the calculations.
Let’s analyze the proof of the square of the sum
(a + b) 2 = a 2 + 2ab + b 2.
The first mathematical regularity
was proved by the ancient Greek scientist Euclid, who worked in Alexandria in the 3rd century BC, he used for this a geometric method of proving the formula, since the scientists of ancient Hellas did not use letters to denote numbers.They commonly used not “a 2”, but “a square on a segment a”, not “ab”, but “a rectangle enclosed between segments a and b”.
Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify a specific person or contact him.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information we collect:
- When you leave a request on the site, we may collect various information, including your name, phone number, email address, etc.
How we use your personal information:
- The personal information we collect allows us to contact you and report unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notices and messages.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, competition or similar promotional event, we may use the information you provide to administer such programs.
Disclosure of information to third parties
We do not disclose information received from you to third parties.
Exceptions:
- If necessary – in accordance with the law, court order, in court proceedings, and / or on the basis of public inquiries or requests from government authorities on the territory of the Russian Federation – disclose your personal information.We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other socially important reasons.
- In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the appropriate third party – the legal successor.
Protecting Personal Information
We take precautions – including administrative, technical and physical – to protect your personal information from loss, theft, and abuse, as well as from unauthorized access, disclosure, alteration and destruction.
Maintaining your privacy at the company level
In order to ensure that your personal information is safe, we communicate the rules of confidentiality and security to our employees, and strictly monitor the implementation of confidentiality measures.
Power formulas
is used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.
Number c
is n
The th power of a
when:
Operations with powers.
1. Multiplying degrees with the same base, their indices add up:
a m
a n = a m + n.
2. In the division of degrees with the same base, their indicators are subtracted:
3. The power of the product of 2 or more factors is equal to the product of the powers of these factors:
(abc …) n = a n b n c n …
4. The power of fraction is equal to the ratio of the powers of the dividend and the divisor:
(a / b) n = a n / b n.
5. Raising the power to the power, the exponents multiply:
(a m) n = a m n.
Each formula above is true from left to right and vice versa.
For example
. (2 · 3 · 5/15) ² = 2² · 3² · 5² / 15² = 900/225 = 4
.
Operations with roots.
1. The root of the product of several factors is equal to the product of the roots of these factors:
2.The root of the ratio is equal to the ratio of the dividend and the divisor of the roots:
3. When raising a root to a power, it is enough to raise the root number to this power:
4. If you increase the degree of the root by n
times and at the same time to build in n
th power of the radical number, then the value of the root will not change:
5. If you decrease the degree of the root by n
times and at the same time extract the root n
th power of the radical number, then the value of the root will not change:
Degree with negative exponent. The power of a number with a non-positive (integer) exponent is defined as a unit divided by the power of the same number with an exponent equal to the absolute value of a non-positive exponent:
Formula a m
: a n = a m – n
can be used not only at m
> n
, but also at m
n
…
For example
. a
4: a 7 = a 4 – 7 = a -3
.
To formula a m
: a n = a m – n
became valid at m = n
, zero degree presence is required.
Degree with zero exponent.
The power of any nonzero number with zero exponent equals one.
For example
. 2 0 = 1, (- 5) 0 = 1, (- 3/5) 0 = 1.
Degree with fractional exponent.
To construct the real number a
to the power of m / n
, you must extract the root n
th degree out of m
The th power of this number is a
.
Formulas or rules of abbreviated multiplication are used in arithmetic, or rather in algebra, for a faster process of calculating large algebraic expressions.The formulas themselves are derived from the rules existing in algebra for multiplying several polynomials.
The use of these formulas provides a fairly quick solution to various mathematical problems, and also helps to simplify expressions. Algebraic transformation rules allow you to perform some manipulations with expressions, following which you can get the expression on the left side of the equality on the right side, or transform the right side of the equality (to get the expression on the left side after the equal sign).
It is convenient to know the formulas used for abbreviated multiplication by memory, as they are often used in solving problems and equations. Below are the main formulas included in this list, and their name.
Sum square
To calculate the square of the sum, you need to find the sum consisting of the square of the first term, twice the product of the first term by the second, and the square of the second. As an expression, this rule is written as follows: (a + c) ² = a² + 2ac + c².
Square of difference
To calculate the square of the difference, you need to calculate the sum consisting of the square of the first number, twice the product of the first number by the second (taken with the opposite sign), and the square of the second number. As an expression, this rule looks as follows: (a – c) ² = a² – 2ac + c².
Difference of squares
The formula for the difference between two numbers squared is equal to the product of the sum of these numbers by their difference.In the form of an expression, this rule looks as follows: a² – c² = (a + c) · (a – c).
Sum cube
To calculate the cube of the sum of two terms, you need to calculate the sum consisting of the cube of the first term, triple product of the square of the first term and the second, triple product of the first term and the second squared, and the cube of the second term. In the form of an expression, this rule looks as follows: (a + c) ³ = a³ + 3a²c + 3ac² + c³.
Cubes sum
According to the formula, it is equated to the product of the sum of these terms by their incomplete square of the difference. In the form of an expression, this rule looks as follows: a³ + c³ = (a + c) · (a² – ac + c²).
Example.
You want to calculate the volume of a shape that is formed by adding two cubes. Only the sizes of their sides are known.
If the side values are small, the calculations are easy.
If the lengths of the sides are expressed in cumbersome numbers, then in this case it is easier to apply the “Sum of Cubes” formula, which will greatly simplify the calculations.
Difference cube
The expression for the cubic difference is as follows: as the sum of the third power of the first term, triple the negative product of the square of the first term by the second, triple the product of the first term by the square of the second, and the negative cube of the second term. In the form of a mathematical expression, the cube of the difference looks like this: (a – c) ³ = a³ – 3a²c + 3ac² – c³.
Cubes difference
The formula for the difference between cubes differs from the sum of cubes in only one sign.Thus, the difference between the cubes is a formula equal to the product of the difference of these numbers by their incomplete square of the sum. In the form of a mathematical expression, the difference between the cubes is as follows: a 3 – c 3 = (a – c) (a 2 + ac + c 2).
Example.
You want to calculate the volume of the shape that will remain after subtracting the yellow volumetric shape, which is also a cube, from the volume of the blue cube. Only the size of the side of the small and large cube is known.
If the side values are small, the calculations are fairly straightforward.And if the lengths of the sides are expressed in significant numbers, then it is worth using a formula entitled “Difference Cubes” (or “Difference Cube”), which will greatly simplify the calculations.
Keywords:
sum square, difference square, sum cube, difference cube, difference
squares, sum of cubes, difference of cubes
- Square of difference
of two quantities is
square of the first minus twice the product of the first plus the second
square of the second.
(a-b) 2 = a 2 -2ab + b 2
- The product of the sum of two values by their difference is equal to of the difference
their squares
.
(a + b) (a-b) = a 2 -b 2
- K
ub sum
of two quantities is equal to a cube
of the first value plus three times the product of the square of the first by the second
plus three times the product of the first by the square of the second plus the cube of the second.(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3
- K
ub difference
of two quantities is equal to the cube of the first
minus the triple product of the square of the first by the second plus triple
product of the first by the square of the second minus the cube of the second.(a-b) 3 = a 3 -3a 2 b + 3ab 2 -b 3
- Product of the sum
two quantities per incomplete square
difference equals sum
their cubes
.
(a + b) (a 2 -ab + b 2) = a 3 + b 3
- Product of the difference of two values by the incomplete
the square of the sum is equal to difference
their cubes.(a – b) (a 2 + ab + b 2) = a 3 –
b 3
Sum square
of two quantities is
square of the first plus twice the product of the first by the second plus
the square of the second value.(a + b) 2 = a 2 + 2ab + b 2
Very often the reduction of a polynomial to the standard form can be done
by applying abbreviated multiplication formulas. They all prove
direct opening of parentheses and reduction of similar terms.
You need to know the abbreviated multiplication formulas by heart:
Example
… Let’s prove
formula a 3 + b 3
= ( a
+ b
) ( a
2 – ab
+ b
2).
We have: ( a
+ b
) ( a
2 – ab
+ b
2) = a
3 – a
2 b
+ ab 2
+ ba
2 – ab
2 – b
3
Bringing similar
terms, we see that
( a
+ b
) ( a
2 – ab
+ b
2)
=
a 3 + b 3
,
which proves the required formula.
It can be proved similarly that
( a
– b
) ( a
2 + ab
+ b
2)
=
a 3 – b 3
It is not enough just to know the abbreviated multiplication formulas by heart.
We must also learn to see this formula in a concrete algebraic expression.
For example:
49m 2 – 42mn
+ 9n 2 = (7m – 3n) 2
Or another more complicated example:
Here 3x 2
can
imagine as (
√
3x) 2
It is also useful to know how to raise a binomial to a power
greater than 3.A formula for writing out the decomposition of an algebraic
sum of two terms of arbitrary degree, was first proposed by Newton in
1664–1665 and was named the Newton binomial. Odds
formulas are called binomial coefficients. If n is positive
an integer, then the coefficients vanish for any k> n,
therefore, the expansion contains only a finite number of terms. In all the rest
cases, the expansion is an infinite (binomial) series.
(The conditions for the convergence of the binomial series were first established at the beginning
19th centuryAbel.) Such special cases as
(a + b) 2 = a 2 + 2ab + b 2
and
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 +
b 3
were known long before Newton. If
n is a positive integer, then the binomial coefficient for a n-k
b k
v
the binomial formula is the number of combinations of n
by k
,
denoted by C k n
… For small
values n
the coefficients can be found from Pascal’s triangle:
in which each of the numbers except ones is equal to
the sum of two adjacent numbers on the line above.For a given n, the corresponding
(nth) row of Pascal’s triangle gives in order the coefficients of the binomial
expansion of the nth degree, which is easy to verify for n = 2 and n = 3.
.