# Exponent Rules Practice — Free Timed Powers and Roots Drill | Quant Daily > Free timed exponent rules practice: evaluate powers, undo squares with roots, and read a worked solution each time. Recall drills, no sign-up, no calculator. Source: https://quantdaily.app/drill-types/exponents-and-roots --- Exponents & roots ## Exponent rules practice — evaluate powers and roots on sight Exponent rules practice pays off fastest when the common powers are recall rather than computation. Squares to 30, cubes to 15 and the powers of 2 to 1024 cover most of what is asked, and a root question is simply the same fact read backwards. When a value is not memorised, split the exponent or use the binomial expansion of a nearby round number. The timed drill here is evaluation: given a power or a root, produce the value. That sounds like pure memory, and in part it is — but the rules are what let you rebuild any fact you have not memorised. Knowing that 9 cubed is 9 squared times 9 turns an unknown into two steps you already have. **Try it against the clock.** Exponents & Roots runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading. [Start a timed Exponents & Roots drill →](https://quantdaily.app/?drill=exponents_roots&src=seo-exponents-and-roots) ### How exponent rules practice turns computation into recall - Split a power using the product rule 9³ is 9² times 9, which is 81 times 9, which is 729. The rule that exponents add on multiplication is what makes a cube reachable from a square you already know. - Read a root as the square you have memorised √784 asks which number squared gives 784. Since 28² is 784, the answer is 28. Roots are not a separate skill; they are the squares table read in the other direction. - Rebuild an awkward square with the binomial expansion 23² equals (20 + 3)², which is 400 plus 120 plus 9, or 529. Anchoring on the nearest multiple of ten makes any two-digit square reachable in a few seconds. - Narrow a root by last digit and magnitude √529 ends in 9, so the root ends in 3 or 7, and 529 sits between 400 and 900, so the root is in the twenties. That leaves 23 and 27, and one check settles it. ### The recall set worth having cold | Know this | Because it gives you | | --- | --- | | Squares to 30² | Every root question up to 900 | | Cubes to 15³ | 9³ = 729, 12³ = 1728 | | Powers of 2 to 2¹⁰ | 1024, and every halving question | | Last digit of a square | Two candidate roots, not ten | | (a + b)² expansion | Any two-digit square from a round anchor | ### 3 sample exponent rules practice, with worked solutions Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers. #### Question 1 Evaluate: √529 Evaluate the root. - 23 - 25 - 24 - 21 - 22 - 23² = 529, so √529 = 23. Anchor on the last digit and the nearest square of a multiple of 10. - Answer: 23. #### Question 2 Evaluate: √576 Evaluate the root. - 24 - 74 - 23 - 25 - 22 - 24² = 576, so √576 = 24. Anchor on the last digit and the nearest square of a multiple of 10. - Answer: 24. #### Question 3 What is the value of 12³? Evaluate the power. - 2197 - 1872 - 1728 - 1740 - 1748 - 12³ = 12² × 12 = 144 × 12 = 1728. (Exam-standard cube — worth memorising to 15³.) - Answer: 1728. ### Common mistakes this drill is built from The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks. #### Multiplying the base by the exponent Reading 9³ as 27 treats the exponent as a multiplier. It means three nines multiplied together, which is 729, and 27 is printed as a distractor. #### Taking only the positive root when both are valid If x² = 529 with no sign restriction, x is 23 or −23. The question usually states x > 0, and when it does not, both matter. #### Guessing a root without the last-digit check A square ending in 9 has a root ending in 3 or 7. Skipping that check turns a two-way decision into a ten-way guess. ### Why your answer came out wrong The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here. #### Is 9³ equal to 27? No — 27 is 3³, and it is also what you get by multiplying 9 by 3. 9³ means 9 × 9 × 9, which is 729. Treating the exponent as a multiplier is the single most common error in this topic, and the resulting value is always an available option. #### If x² = 529, is x necessarily 23? Only if the question restricts x to positive values, which it usually does. Without that restriction, both 23 and −23 satisfy the equation. Read the constraint line before answering — it is there precisely because both roots are valid otherwise. #### Is √784 close to 26? It is exactly 28. The last digit 4 means the root ends in 2 or 8, and 784 sits just under 800, so the root is close to 28 rather than 26. Estimating without the last-digit check is what produces near-misses like this, and the near-miss is on the option list. ### FAQ #### What is the fastest way to evaluate a cube like 9³? Split it using the product rule: 9³ is 9² times 9, so 81 times 9, which is 729. Any cube you have not memorised is reachable from its square in one multiplication, which is far faster than multiplying the base three times. #### How do I find a square root without a calculator? Use the last digit and the magnitude together. A number ending in 9 has a root ending in 3 or 7, and comparing against the nearest hundreds squares fixes the tens digit. For 529, the root is in the twenties and ends in 3 or 7, so it is 23 or 27 — and 23² is 529. #### Which powers are genuinely worth memorising? Squares up to 30, cubes up to 15, and the powers of 2 up to 1024. That set covers the overwhelming majority of what gets asked, and everything else can be rebuilt from it with the product rule or a binomial expansion in a couple of seconds. ### Practise the next topic - [All 51 drill types](https://quantdaily.app/drill-types) — the shared daily cycles 36 of 51, with every published time limit. - [Square root and cube root questions — read the answer off the last digit](https://quantdaily.app/drill-types/square-root-cube-root) — square root and cube root questions, timed. - [How to square numbers fast — two shortcuts cover almost everything](https://quantdaily.app/drill-types/square-shortcuts) — how to square numbers fast, timed. - [Speed tricks](https://quantdaily.app/drill-types) — the seven mental-math shortcuts that make a 15-second answer possible. - [How to improve calculation speed](https://quantdaily.app/calculation-speed) — the six-week arc these drills fit into. Quant Daily is free, needs no account, and is not affiliated with ETS or GMAC. GRE/GMAT-style practice only.