Square Root and Cube Root Tricks: The Short Answer
For a perfect square or perfect cube, you never need long division. Split the number, read the unit digit of the root from the last digit, and read the tens part from the leading digits. For example, √7744 = 88 and ∛103823 = 47, each in two steps.
Key takeaway for SSC: Learn squares up to 30 and cubes up to 20, plus the two unit-digit tables below. That is enough for almost every root inside a simplification question. Every number on this page was checked with a program.
Roots appear inside simplification and number system questions in SSC CGL, SSC CHSL, SSC MTS, SSC GD and SSC CPO, such as "find √1764 + ∛2744 − √196". For powers and surds, see surds and indices for SSC CGL; for last-digit questions, see unit digit tricks for SSC CGL.
Squares from 1 to 30
| n | n² | n | n² | n | n² |
|---|---|---|---|---|---|
| 1 | 1 | 11 | 121 | 21 | 441 |
| 2 | 4 | 12 | 144 | 22 | 484 |
| 3 | 9 | 13 | 169 | 23 | 529 |
| 4 | 16 | 14 | 196 | 24 | 576 |
| 5 | 25 | 15 | 225 | 25 | 625 |
| 6 | 36 | 16 | 256 | 26 | 676 |
| 7 | 49 | 17 | 289 | 27 | 729 |
| 8 | 64 | 18 | 324 | 28 | 784 |
| 9 | 81 | 19 | 361 | 29 | 841 |
| 10 | 100 | 20 | 400 | 30 | 900 |
Cubes from 1 to 20
| n | n³ | n | n³ |
|---|---|---|---|
| 1 | 1 | 11 | 1331 |
| 2 | 8 | 12 | 1728 |
| 3 | 27 | 13 | 2197 |
| 4 | 64 | 14 | 2744 |
| 5 | 125 | 15 | 3375 |
| 6 | 216 | 16 | 4096 |
| 7 | 343 | 17 | 4913 |
| 8 | 512 | 18 | 5832 |
| 9 | 729 | 19 | 6859 |
| 10 | 1000 | 20 | 8000 |
Unit Digit Tables for Roots
| Last digit of the square | Unit digit of the square root |
|---|---|
| 1 | 1 or 9 |
| 4 | 2 or 8 |
| 5 | 5 |
| 6 | 4 or 6 |
| 9 | 3 or 7 |
| 0 (even number of zeros) | 0 |
A perfect square never ends in 2, 3, 7 or 8, and never ends in an odd number of zeros.
| Last digit of the cube | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Unit digit of the cube root | 0 | 1 | 8 | 7 | 4 | 5 | 6 | 3 | 2 | 9 |
For cubes, every last digit gives exactly one root digit: 0, 1, 4, 5, 6 and 9 stay the same, while 2 ↔ 8 and 3 ↔ 7 swap.
How to Find the Square Root of a Perfect Square (Unit Digit Method)
- Remove the last two digits. Find the largest number a whose square is not more than what is left.
- From the last digit, list the two possible unit digits of the root.
- Compare the number with (a5)² = a × (a + 1) × 100 + 25. If the number is bigger, take the larger unit digit; if smaller, take the smaller one.
| Number | Left part → a | Possible roots | (a5)² | Square root |
|---|---|---|---|---|
| 7744 | 77 → 8 | 82 or 88 | 85² = 7225 | 88 |
| 5329 | 53 → 7 | 73 or 77 | 75² = 5625 | 73 |
| 9216 | 92 → 9 | 94 or 96 | 95² = 9025 | 96 |
| 15129 | 151 → 12 | 123 or 127 | 125² = 15625 | 123 |
Shortcut for squares ending in 5: (a5)² = a × (a + 1), followed by 25. So 45² = 4 × 5 = 20 → 2025, and √2025 = 45.
How to Find the Cube Root of a Perfect Cube
- Remove the last three digits. Find the largest a whose cube is not more than what is left.
- Read the unit digit of the root from the cube table above.
| Number | Left part → a | Last digit → unit digit | Cube root |
|---|---|---|---|
| 103823 | 103 → 4 (64 ≤ 103 < 125) | 3 → 7 | 47 |
| 474552 | 474 → 7 (343 ≤ 474 < 512) | 2 → 8 | 78 |
| 857375 | 857 → 9 (729 ≤ 857 < 1000) | 5 → 5 | 95 |
| 39304 | 39 → 3 (27 ≤ 39 < 64) | 4 → 4 | 34 |
Square Root and Cube Root by Prime Factorisation
Pair the prime factors for a square root and make triplets for a cube root.
| Number | Prime factors | Root |
|---|---|---|
| 1764 | 2² × 3² × 7² | √1764 = 2 × 3 × 7 = 42 |
| 3136 | 2⁶ × 7² | √3136 = 2³ × 7 = 56 |
| 9261 | 3³ × 7³ | ∛9261 = 3 × 7 = 21 |
| 2744 | 2³ × 7³ | ∛2744 = 2 × 7 = 14 |
Smallest multiplier questions
- Smallest number to multiply 1176 by to get a perfect square: 1176 = 2³ × 3 × 7². The unpaired factors are one 2 and one 3, so multiply by 6. 1176 × 6 = 7056 = 84².
- Smallest number to multiply 675 by to get a perfect cube: 675 = 3³ × 5². One more 5 completes the triplet, so multiply by 5. 675 × 5 = 3375 = 15³.
Square Roots and Cube Roots of Decimals
Count decimal places: a square root halves them and a cube root divides them by three.
| Expression | Working | Answer |
|---|---|---|
| √18.49 | √1849 = 43, 2 places → 1 place | 4.3 |
| √0.0049 | √49 = 7, 4 places → 2 places | 0.07 |
| √0.0169 | √169 = 13, 4 places → 2 places | 0.13 |
| ∛0.000343 | ∛343 = 7, 6 places → 2 places | 0.07 |
Quick Checks Before You Mark an Option
- Digit sum test: the digit sum (repeated until one digit) of a perfect square is always 1, 4, 7 or 9. If an option's square has digit sum 2, 3, 5, 6 or 8, it cannot be a perfect square.
- Last digit test: a number ending in 2, 3, 7 or 8 is not a perfect square.
- Size test: 30² = 900, 100² = 10,000 and 20³ = 8000; place the root between two round numbers before choosing.
Common Mistakes with Roots
- Choosing the wrong one of the two possible unit digits (always compare with (a5)²).
- Splitting off two digits for a cube root; a cube root needs the last three digits removed.
- Forgetting that 2 ↔ 8 and 3 ↔ 7 swap for cube roots but not for square roots.
- Miscounting decimal places in √0.0049 (the answer is 0.07, not 0.7).
Square Root and Cube Root Practice Questions with Answers
- Find √6889. Answer: 83. 68 → 8; last digit 9 → 83 or 87; 85² = 7225 is bigger than 6889, so 83.
- Find √12544. Answer: 112. 125 → 11; last digit 4 → 112 or 118; 115² = 13225 is bigger, so 112.
- Find ∛29791. Answer: 31. 29 → 3; last digit 1 → 1.
- Find ∛658503. Answer: 87. 658 → 8 (512 ≤ 658 < 729); last digit 3 → 7.
- Find the value of √1764 + ∛2744 − √196. Answer: 42. 42 + 14 − 14 = 42.
- What is the smallest number by which 1176 must be multiplied to make it a perfect square? Answer: 6. 1176 × 6 = 7056 = 84².
For timed practice, take the SSC MTS Number System Mock Test 03 and the SSC GD Number System Mock Test 03.
Square Root and Cube Root FAQs
How do you find the square root of a 4-digit number quickly?
Remove the last two digits, find the largest square below the remaining number, list the two unit digits allowed by the last digit, and compare with (a5)² to choose one. √7744 = 88 this way.
What is the trick for cube roots?
Remove the last three digits and find the largest cube below what is left for the tens digit. The unit digit comes from the last digit: 2 ↔ 8 and 3 ↔ 7 swap, the rest stay the same.
Can a perfect square end in 7?
No. A perfect square can end only in 0, 1, 4, 5, 6 or 9.
How many squares and cubes should I learn for SSC?
Squares up to 30 and cubes up to 20 cover most SSC simplification questions; the methods on this page handle larger perfect squares and cubes.
Practice Next
- Topic tests: SSC MTS Number System Mock Test 03, SSC GD Number System Mock Test 03, SSC CGL Number System Mock Test 06, SSC MTS Simplification Mock Test 01 and SSC CHSL Simplification Mock Test 01.
- Related notes: number system for SSC CGL, surds and indices for SSC CGL and unit digit tricks for SSC CGL.
- Browse all study notes.