TMUA Surds and Square Roots

It is vital to be comfortable manipulating surds, which appear in all sorts of TMUA questions - we also include here square root questions which invariably involve surds and are another TMUA favourite. Note: There are quite a few traps to fall into involving square roots, and TMUA loves setting them.

For those so interested, we offer both one to one TMUA coaching on surds and square roots, as well as online group coaching via Zoom - 'TMUA Prerequisite Maths One' is Session #1 of 20' . 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 surds and square roots - 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.

• Teaser #289 - tests ability to find a geometric series with surd terms given clues - Fairly Mild
• Teaser #440 - tests ability to simplify powers within a surd - Standard
• Teaser #281 - tests how to evaluate both trig values and manipulate surds mentally - Standard
• Teaser #243 - tests ability to manipulate surd expression to high power - Standard
• Teaser #378 - tests finding the distance between two points and requires surd manipulation - Standard
• Teaser #3 - tests how to find furthest distance of a circle from the origin - Standard
• Teaser #376 - tests comparing surd coefficient of two cubics to find unknowns - Standard
• Teaser #181 - tests how to solve an exponential equation with surds as the exponent - Standard
• Teaser #11 - tests ability to evaluate 0.25^0.25 - Standard
• Teaser #99 - tests ability to apply Heron's Formula to a triangle word problem (the final calculation is beyond TMUA) - Standard
• Teaser #287 - tests knowledge of sin and cos double angles and involves surd manipulation - Standard
• Teaser #136 - tests how to find the convergence value of a sequence - Standard
• Teaser #229 - tests ability to manipulate powers of surds using difference of two squares - Standard
• Teaser #129 - tests finding roots of a quartic given two roots, one of which is a surd - Standard
• Teaser #41 - tests finding area of triangle given rhombus and square - Standard
• Weekend Workout #1 - tests ability to evaluate a power tower of fraction exponents - Standard
• Teaser #193 - tests application of double angle formulae twice, involves surd manipulation - Standard
• Teaser #121 - tests how to find the radius of a reduced sphere - Standard
• Teaser #406 - tests basic manipulation of larger surds into smaller surds - 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 #231 - tests ability to simplify a surd expression to a high power - Medium
• Teaser #143 - tests how to find the maximum area of a rectangle inscribed in a semicircle - Medium
• Weekend Workout #3 - tests ability to manipulate surds under square root by noticing a difference of squares - Medium
• Teaser #500 - tests understanding of when sqrt(x^2) = (sqrt(x))^2 - Medium
• Teaser #1 - tests how to simplify a square root under a square root - Medium
• Teaser #427 - tests ability to arrange various expressions in increasing order by clever approximations - Medium
• Teaser #153 - tests how to solve an equation involving cos(sin(x)) involving square roots - Medium
• Teaser #257 - tests ability to calculate the value of a power tower of surds - Medium
• Teaser #245 - tests ability to disentangle a plethora of powers within powers - Medium
• Teaser #272 - tests knowledge of domains of both logs and square roots - Medium
• Teaser #203 - tests how to find the radius of a circle inscribed in an equilateral triangle with surd results - Medium
• Teaser #28 - tests understanding of power rules to simplify a complex expression - Medium
• Teaser #185 - tests manipulation of position vectors to see when two particles reach their closest point which involves finding the minimum of a square root - Medium
• Teaser #190 - tests ability to arrange surds in increasing order using a little known inequality - Medium
• Teaser #215 - tests ability to expand a cubic and compare surd and non surd coefficients - Medium
• Weekend Workout #16 - tests ability to see a hidden quadratic under a fourth root and then 'completing the square' to find a minimum  - Medium
• Weekend Workout #15 - tests manipulation of powers of surds and comparison of the results - Medium
• Teaser #494 - tests understanding of how to find positive solutions to a modulus equation - Medium
• Teaser #227 - tests ability to manipulate surds in exponent - Medium
• Teaser #298 - tests ability to differentiate and compare results using surd manipulation - Quite Tough
• Teaser #504 - tests advanced manipulation of two log(log) functions to solve an inequality - Quite Tough
• Teaser #465 - tests ability to extricate a square root from under a square root then cube it - Quite Tough
• Teaser #247 - tests ability to manipulate surds in exponent (probably too hard for TMUA but worth knowing) - Quite Tough
• Weekend Workout #19 - tests ability to manipulate surds in exponent (probably too hard for TMUA but worth knowing) - Tough

TMUA Past Questions involving Surds and Square Roots: SP p1 q12; SP p1 q19; SP p2 q1; SP p2 q2; 2016 p1 q1; 2016 p1 q16; 2016 p1 q19; 2016 p1 q20; 2016 p2 q2; 2017 p2 q3; 2017 p2 q8; 2018 p1 q7; 2018 p1 q15; 2018 p1 q16; 2018 p2 q9; 2018 p2 q20; 2019 p1 q4; 2019 p1 q9; 2019 p1 q14; 2019 p2 q19; 2020 p1 q1; 2020 p1 q9; 2020 p1 q12; 2021 p2 q15; 2021 p2 q17; 2021 p2 q19; 2022 p1 q13; 2022 p1 q14; 2022 p2 q12; 2023 p1 q10; 2023 p2 q1; 2023 p2 q2; 2023 p2 q3

Practice makes perfect - good luck in your studies!