Unit-digit table • Square roots by grouping • Cube root of perfect cube trick • Decimals • Simplifying surds
Exam questions on roots almost always use exact squares and cubes, because the setter wants a clean answer. That lets you find the root from just two pieces of information: the last digit of the number, which fixes the last digit of the root, and the leading digits, which fix the rest. You never need long division.
Learn the squares of 1 to 30 and the cubes of 1 to 15 if you have not already. Every method on this page compares the leading part of the number with those values.
| Number ends in | Square root ends in | Cube root ends in |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 or 9 | 1 |
| 2 | not a square | 8 |
| 3 | not a square | 7 |
| 4 | 2 or 8 | 4 |
| 5 | 5 | 5 |
| 6 | 4 or 6 | 6 |
| 7 | not a square | 3 |
| 8 | not a square | 2 |
| 9 | 3 or 7 | 9 |
Two things stand out. Squares give two possible endings, so you need one extra step to choose. Cubes give exactly one ending for every digit, which is why cube roots of perfect cubes are even faster than square roots. For cubes, 2 and 8 swap, 3 and 7 swap, and every other digit stays the same.
Step 3 works because (10a + 5)² = 100 × a(a + 1) + 25, so a(a + 1) is the dividing line between the smaller and larger choices.
The last digit 6 gives an ending of 4 or 6. What is left is 57, and 7² = 49 is the largest square not above it, so the tens digit is 7. Since 57 > 7 × 8 = 56, take the larger ending: 76.
The ending is 3 or 7. What is left is 18, and 4² = 16 gives tens digit 4. Since 18 < 4 × 5 = 20, take the smaller ending: 43.
The ending is 2 or 8. What is left is 23, with tens digit 4. Since 23 > 20, the root is 48.
There is no third step, because each ending is unique.
It ends in 2, so the root ends in 8. What is left is 474, and 7³ = 343 is the largest cube not above it. The root is 78.
It ends in 3, so the root ends in 7. What is left is 19, and 2³ = 8 is the largest cube not above it. The root is 27.
1,350 = 2 × 3³ × 5². Each prime's exponent must be a multiple of 3, so multiply by 2² × 5 = 20. Then 1,350 × 20 = 27,000, whose cube root is 30.
For decimals, count the decimal places. A square root halves them, so the count must be even; a cube root divides them by three. Find the root of the digits and put the point back.
Four decimal places become two. √169 = 13, so the answer is 0.13.
Likewise ∛0.000343 = 0.07, because six decimal places become two and ∛343 = 7.
When the number is not a perfect square, simplify the surd by taking out the largest square factor. Only like surds can be added or subtracted.
72 = 36 × 2, so √72 = 6√2. Check: 6√2 is about 8.49, and 8.49² is close to 72.
Prime factorisation, used above to make perfect cubes, is the same tool behind the number of factors formula. The unit-digit cycles explain the table above, and more shortcuts are in the Quant cheat code.
Q1. What is the square root of 7,744? easy
✅ Answer: (b) 88
💡 Explanation: The number ends in 4, so the root ends in 2 or 8. 8² = 64 is the largest square not above 77, giving tens digit 8. Since 77 > 8 × 9 = 72, take the larger ending: 88.
Q2. What is the cube root of 205,379? easy
✅ Answer: (a) 59
💡 Explanation: A cube ending in 9 has a root ending in 9. Drop the last three digits to get 205; 5³ = 125 is the largest cube not above it, so the root is 59.
Q3. What is the square root of 0.0729? medium
✅ Answer: (c) 0.27
💡 Explanation: 0.0729 has four decimal places, an even number, so its square root has two. √729 = 27, so √0.0729 = 0.27.
Q4. Which of these numbers cannot be a perfect square? easy
✅ Answer: (c) 8,463
💡 Explanation: No perfect square ends in 2, 3, 7 or 8, so 8,463 is ruled out at once. The others are 74², 79² and 96².
Q5. Simplify √50 + √18. medium
✅ Answer: (d) 8√2
💡 Explanation: √50 = √(25 × 2) = 5√2 and √18 = √(9 × 2) = 3√2. Like surds add like terms, so the sum is 8√2.
Q6. By what smallest number must 1,350 be multiplied to make it a perfect cube? hard
✅ Answer: (a) 20
💡 Explanation: 1,350 = 2 × 3³ × 5². To make every exponent a multiple of 3, multiply by 2² × 5 = 20. The product is 27,000 = 30³.
No. The unit-digit step assumes the number is an exact square or cube. For other numbers, bracket the root between two perfect squares or cubes and estimate, or simplify the surd by taking out square factors.
The same steps apply. Drop the last two digits and find the largest square not above what remains. For 6-digit squares that remaining part has up to four digits, so the tens part of the root can be two digits long.
All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.
How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 14 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-20 against the sources listed above.
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