TMUA Exponents and Power Rules

TMUA occasionally tests this topic directly, mainly in the form of exponential equations that invariable lead to a quadratic. They also frequently test exponents directly together with logarithms. Indirectly, exponents and power rules are as essential to success in TMUA as general algebra and must be thoroughly understood.

For those so interested, we offer both one to one TMUA coaching on Exponents and Power Rules, as well as online group coaching via Zoom - 'TMUA Prerequisite Maths One' is Session #1 of 20 and covers the basic rules, and 'TMUA Exponents and Logarithms' is Session #3 of 20 and covers what needs knowing for TMUA success. Check our TMUA Group Coaching Schedule page for dates when these sessions are available - details of what we cover in each group session can be found on our TMUA Coaching Sessions Overview page. For more general information on our online sessions check out our  'Online Coaching FAQ'.

Feel free to contact us if you require more information regarding any of the above.

Free Revision Resources

The below list of questions links to short videos (teasers) on our YouTube channel which directly or indirectly test elements of what is required to succeed in answering TMUA questions on exponents and power rules - they also provide a detailed solution to the question posed, together with tips and tricks and are arranged roughly in increasing order of difficulty. All the videos are taken from the Gresty Academy YouTube podcast 'A Crash Course in TMUA Must Know Maths' which contains hundreds of TMUA style questions and is great for revision.

• Weekend Workout #12 - tests ability to manipulate a numerical expression involving powers using difference of two squares - Fairly Mild
• Weekend Workout #6 - tests general ability to manipulate percentages and fractions of powers - Fairly Mild
• Teaser #178 - tests ability to find a pattern to simplify and sum a series of powers - Fairly Mild
• Teaser #316 - tests ability to raise to positive and negative powers - Fairly Mild
• Teaser #292 - tests ability to deal with negatives inside and outside of brackets per BIDMAS, whilst squaring - Fairly Mild
• Teaser #440 - tests ability to simplify powers within a surd - Standard
• Teaser #423 - tests how to find the number of perfect square factors (ie products of even powers) of 162,000 - Standard
• Teaser #493 - tests understanding of the various components of logarithms and exponents and their allowed values - Standard
• Weekend Workout #1 - tests ability to evaluate a power tower of fraction exponents - Standard
• Teaser #53 - tests how to sum a hidden geometric series - Standard
• Teaser #181 - tests how to solve an exponential equation with surds as the exponent - Standard
• Teaser #102 - tests ability to simplify a sum of large powers of 2 - Standard
• Teaser #110 - tests understanding of percentage increase and decrease - Standard
• Teaser #174 - tests ability to solve an exponential equation using manipulation - Standard
• Teaser #94 - tests understanding of percentage increases (calculation too hard for TMUA, method useful) - Standard
• Teaser #11 - tests ability to evaluate 0.25^0.25 - Standard
• Teaser #234 - tests ability to evaluate mentally a power tower - Standard
• Teaser #243 - tests ability to manipulate surd expression to high power - Standard
• Weekend Workout #21 - tests ability to correctly evaluate a power of a surd expression - Standard
• Weekend Workout #22 - tests ability to correctly evaluate a power tower of surds - Standard
• Teaser #229 - tests ability to manipulate powers of surds using difference of two squares - Standard
• Teaser #427 - tests ability to arrange various expressions in increasing order by clever approximations - Medium
• Teaser #472 - tests ability to solve a trio of log equations with unknown bases - Medium
• Teaser #28 - tests understanding of power rules to simplify a complex expression - Medium
• Teaser #126 - tests how to find the equation of the gradient of a fairly complex function at the origin - Medium
• Teaser #501 - tests understanding of a well known functional inequality - Medium
• Teaser #500 - tests understanding of when sqrt(x^2) = (sqrt(x))^2 - Medium
• Teaser #476 - tests how to solve an inequality involving a polynomial over a polynomial - Medium
• Teaser #161 - tests ability to evaluate a fraction involving power towers - Medium
• Teaser #257 - tests ability to calculate the value of a power tower of surds - Medium
• Teaser #197 - tests ability to find number of solutions to e^(2x)+x-3=0  in an interval- Medium
• Teaser #426 - tests understanding of how to solve an exponential equation with x being a power of a power of a power - Medium
• Teaser #478 - tests ability to find the maximum of an exponential expression - Medium
• Teaser #225 - tests ability to evaluate a power tower including fractions - Medium
• Teaser #231 - tests ability to simplify a surd expression to a high power - Medium
• Teaser #488 - tests understanding of log bases and their respective graphs - Medium
• Teaser #245 - tests ability to disentangle a plethora of powers within powers - Medium
• Teaser #260 - tests ability to arrange fractional powers from smallest to largest using a neat trick - Medium
• Teaser #505 - tests ability to find the coefficient where powers of a and b are the same in a binomial expansion - Medium
• Teaser #274 - tests how to find f(f(f(f(f(x))))) where f(x) involves powers and roots by spotting a pattern - Medium
• Teaser #300 - tests understanding of how to evaluate power towers depending on the position of brackets - Medium
• Teaser #90 - tests understanding of compound interest calculations - (the calculation itself is beyond TMUA) - Medium
• Teaser #490 - tests ability to find a maximum of an exponential trigonometric expression - Medium
• Weekend Workout #15 - tests manipulation of powers of surds and comparison of the results - Medium
• Weekend Workout #9 - tests how to solve an exponential equation of form a^x=b^y using 'change of base of exponents'- Medium
• Teaser #227 - tests ability to manipulate surds in exponent - Medium
• Teaser #443 - tests how to find a specific number of perfect square factors of a number - Quite Tough
• Weekend Workout #4 - tests knowledge that e^(ln(a))=a - Quite Tough
• Teaser #198 - tests ability to simplify ln(ln(x))=e - Quite Tough
• Teaser #322 - tests finding a pattern to sum a series of powers of a root of a cubic - Quite Tough
• Teaser #298 - tests ability to differentiate and compare results involving exponent manipulation - Quite Tough
• Weekend Workout #10 - tests knowledge of using log rules to simplify an exponential equation - Quite Tough
• Teaser #247 - tests ability to manipulate surds in exponent (probably too hard for TMUA but worth knowing) - Quite Tough
• Teaser #504 - tests advanced manipulation of two log(log) functions to solve an inequality - Quite Tough
• Teaser #282 - tests noticing even and odd functionality to massively simplify an integral - Quite Tough
• Teaser #465 - tests ability to extricate a square root from under a square root then cube it - Quite Tough
• Weekend Workout #19 - tests ability to manipulate surds in exponent (probably too hard for TMUA but worth knowing) - Tough

[Note: We also have an 'Exponential Equations/Power Rules' playlist on our YouTube channel however the videos in this playlist are not all relevant to TMUA (some are too easy, some are too hard, and many cover topics not tested in TMUA).]

TMUA Past Questions involving Exponents and Power Rules: SP p1 q5; SP p1 q8; SP p1 q11; SP p1 q14; SP p1 q16; SP p2 q5; SP p2 q11; 2016 p1 q11; 2016 p1 q18; 2016 p2 q2; 2016 p2 q6; 2016 p2 q7; 2016 p2 q10; ; 2017 p1 q2; 2017 p1 q14; 2017 p2 q1; 2017 p2 q6; 2017 p2 q7; 2017 p2 q13; 2017 p2 q18; 2018 p1 q15; 2018 p2 q1; 2018 p2 q11; 2018 p2 q19; 2019 p1 q3; 2019 p1 q4; 2019 p1 q9; 2019 p1 q12; 2019 p1 q13; 2019 p1 q15; 2020 p1 q1; 2020 p1 q6; 2020 p1 q7; 2020 p1 q13; 2020 p2 q5; 2021 p1 q4; 2021 p1 q11; 2021 p1 q19; 2021 p2 q1; 2021 p2 q14; 2022 p1 q6; 2022 p1 q8; 2022 p1 q13; 2022 p2 q12; 2022 p2 q15; 2022 p2 q18; 2023 p1 q7; 2023 p1 q12; 2023 p1 q15; 2023 p2 q3; 2023 p2 q6; 2023 p2 q11; 2023 p2 q17

Practice makes perfect - good luck in your studies!