What to memorise
- Squares: 1 to 30. For example 13² = 169, 17² = 289, 23² = 529, 29² = 841.
- Cubes: 1 to 15. For example 6³ = 216, 7³ = 343, 9³ = 729, 12³ = 1,728, 15³ = 3,375.
- Last digits of squares: a perfect square ends in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8.
Shortcuts for squares
- Numbers ending in 5: n5² = n(n + 1) followed by 25. So 65² = 6 × 7 = 42, then 25: 4,225.
- Near a base of 10, 50 or 100: 98² = (100 − 2)² = 9,604; 52² = (50 + 2)² = 2,704.
- Difference of squares: 63² − 57² = 120 × 6 = 720.
- Sum of the first n odd numbers is n².
Square roots
Perfect squares by estimation. To find √1,764: it lies between 40² = 1,600 and 50² = 2,500, so the tens digit is 4. The last digit is 4, so the root ends in 2 or 8. Since 1,764 is closer to 1,600 than to 2,500 (and 45² = 2,025 is already above it), the root is 42. Check: 42² = 1,764 ✓.
Non-perfect squares. √200 ≈ 14.14 because 14² = 196 and 15² = 225; 200 is 4 above 196, so add 4 ÷ (2 × 14) = 0.14.
Cube roots by last digit
The last digit of a cube identifies the last digit of the root: 1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9, 0→0. For ∛2,744: it lies between 10³ = 1,000 and 20³ = 8,000, so the tens digit is 1; the last digit 4 gives 4; the root is 14. Check: 14³ = 2,744 ✓.
Worked examples
Example 1. 47² = (50 − 3)² = 2,500 − 300 + 9 = 2,209.
Example 2. 85² = 8 × 9 = 72, then 25: 7,225.
Example 3. √5,625: it lies between 70² = 4,900 and 80² = 6,400, so the tens digit is 7; it ends in 5 so the root is 75. Check: 7 × 8 = 56, then 25: 5,625 ✓.
Example 4. 1,001² = 1,002,001.
Example 5. Is 2,023 a perfect square? It ends in 3, so no.
Common traps
- Using the last-digit rule for cubes on squares.
- Guessing the tens digit without bounding the root.
- Arithmetic slips in expansions such as (a − b)².
A daily drill (10 minutes)
- Two minutes: recite or write squares 1 to 30 from memory.
- Three minutes: ten random squares using the shortcuts.
- Three minutes: five square roots and cube roots by the last-digit and bounding method.
- Two minutes: note the ones you missed.
Where it appears
These skills speed up simplification, mensuration and data interpretation, and appear directly in SSC CGL, banking and railway mathematics. See mental maths techniques for more.
Practice tests below are published by institutes and educators on TestMint.
