1. Must-know values
Squares 1–30 and cubes 1–20. Key ones: 13² = 169, 17² = 289, 19² = 361, 23² = 529, 27² = 729; 12³ = 1728, 13³ = 2197, 14³ = 2744, 16³ = 4096, 18³ = 5832.
2. Square-root by unit digit + nearest square
| Square ends in | Root ends in |
|---|---|
| 1 | 1 or 9 |
| 4 | 2 or 8 |
| 5 | 5 |
| 6 | 4 or 6 |
| 9 | 3 or 7 |
| 00 | 0 |
A perfect square never ends in 2, 3, 7, 8 or an odd number of zeros. Its digital sum is always 1, 4, 7 or 9.
√7056: 80² = 6400, 85² = 7225 → between 80 and 85, ends in 6 → 84 (84² = 7056).
3. Cube roots
Unit digits of cubes: 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9, 0→0. Drop the last three digits; the cube just below the rest gives the tens digit.
∛39304: unit 4 → root ends 4; 39 lies between 27 (3³) and 64 → tens digit 3 → 34.
4. Decimals and fractions
- Decimal places in the root = half those in the square: √0.0144 = 0.12; √1.5625 = 1.25.
- √(a/b) = √a/√b: √(169/225) = 13/15.
5. Making a perfect square
- Least number to subtract: N − (largest square ≤ N).
- Least number to add: (smallest square ≥ N) − N.
- Multiply/divide: factorise; the prime with an odd power is the answer.
(ASO 2025) 1780 + ? = square: 43² = 1849 → 69.
Smallest multiplier for 1323 = 3³ × 7² → multiply by 3 (to get 3⁴ × 7² = 63²).
6. Speed tricks
- Numbers near 50: 48² = 2500 − 200 + 4 = 2304.
- Ending in 5: 65² → 6 × 7 = 42, then 25 → 4225.
- Number of digits in √N: (n + 1)/2 for odd n, n/2 for even n.
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
Basic level
Example 1. √69,169 = ?
(a) 261 (b) 263 (c) 264 (d) 273
Answer: (b) 263
Solution: 263 × 263 = 69,169.
Example 2. ∛3,14,432 = ?
(a) 68 (b) 71 (c) 69 (d) 67
Answer: (a) 68
Solution: 68³ = 3,14,432.
ASO exam level
Example 3. √2.25 = ?
(a) 1.4 (b) 1.7 (c) 1.65 (d) 1.5
Answer: (d) 1.5
Solution: 1.5² = 2.25. So √2.25 = 1.5.
Example 4. √5,184 = ?
(a) 72 (b) 76 (c) 70 (d) 74
Answer: (a) 72
Solution: 72² = 5,184. Tip: last digit and nearest known square narrow it down quickly.
Example 5. √1,764 = ?
(a) 40 (b) 44 (c) 46 (d) 42
Answer: (d) 42
Solution: 42² = 1,764. Tip: last digit and nearest known square narrow it down quickly.
Example 6. The cube root of 15,625 is:
(a) 26 (b) 25 (c) 28 (d) 24
Answer: (b) 25
Solution: 25³ = 15,625.
Example 7. The cube root of 3,375 is:
(a) 18 (b) 16 (c) 14 (d) 15
Answer: (d) 15
Solution: 15³ = 3,375.
Example 8. The least number to be subtracted from 77,765 to make it a perfect square is:
(a) 76 (b) 481 (c) 482 (d) 480
Answer: (b) 481
Solution: 278² = 77,284 ≤ 77,765 < 279². Subtract 77,765 − 77,284 = 481.
Example 9. The least number to be subtracted from 23,472 to make it a perfect square is:
(a) 64 (b) 244 (c) 62 (d) 63
Answer: (d) 63
Solution: 153² = 23,409 ≤ 23,472 < 154². Subtract 23,472 − 23,409 = 63.
Example 10. √60.84 = ?
(a) 7.9 (b) 78.0 (c) 7.8 (d) 0.78
Answer: (c) 7.8
Solution: 7.8² = 60.84. Decimal places in the root are half those in the square.
Example 11. √0.0784 = ?
(a) 0.29 (b) 0.028 (c) 2.80 (d) 0.28
Answer: (d) 0.28
Solution: 0.28² = 0.0784. Decimal places in the root are half those in the square.
Advanced level
Example 12. The smallest number by which 1,200 must be multiplied to make it a perfect square is:
(a) 4 (b) 20 (c) 6 (d) 3
Answer: (d) 3
Solution: 1,200 = 3 × 20². Multiply by 3 to get 3,600 = (60)².
Topic-wise practice for Square roots and cube roots (Quant): Finding roots quickly, perfect squares. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Square root by factorisation and division
- Cube root by factorisation
- Least number to multiply or divide to make a perfect square or cube
- Square roots of decimals and fractions
- Number of digits in a square root