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🟦 Squares & Square Roots: Fast Tricks
Squares appear in area, Pythagoras, algebra and nearly every Class 8 exam. Knowing squares up to 30 by heart and a few tricks for the rest makes calculation much faster.
Squares to know
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900.
Trick 1 — numbers ending in 5
Multiply the tens part by the next number and write 25 after it. 35² → 3 × 4 = 12 → 1225. 85² → 8 × 9 = 72 → 7225.
Trick 2 — numbers near 50
For 50 + d: (25 + d) hundreds, plus d². 53² → 25 + 3 = 28 → 2800 + 9 = 2809. 47² → 25 − 3 = 22 → 2200 + 9 = 2209.
Trick 3 — numbers near 100
96² is 4 short of 100: subtract 4 → 92, then 4² = 16 → 9216. 104²: add 4 → 108, then 16 → 10816.
Trick 4 — use (a + b)²
(a + b)² = a² + 2ab + b². 42² = 40² + 2 × 40 × 2 + 2² = 1600 + 160 + 4 = 1764.
Square roots of perfect squares
To find √5329:
- Last digit: a square ending in 9 comes from a number ending in 3 or 7.
- Leading part: ignore the last two digits → 53. The largest square below 53 is 49 = 7², so the root is in the 70s.
- Choose: 75² = 5625 is more than 5329, so take the smaller option → 73. (Check: 73² = 5329 ✔)
| Square ends in | Root ends in |
|---|---|
| 1 | 1 or 9 |
| 4 | 2 or 8 |
| 5 | 5 |
| 6 | 4 or 6 |
| 9 | 3 or 7 |
| 0 | 0 |
A perfect square never ends in 2, 3, 7 or 8 — a quick way to rule out wrong options in MCQs.
Practise these with instant checking in BeadBop's Mind Tricks and the Class 8 topics in Olympiad Prep.
Quick answers
How do you square a number ending in 5 quickly?
Multiply the tens digit by the next whole number and write 25 after it. For 65²: 6 × 7 = 42, so 65² = 4225.
Can a perfect square end in 7?
No. Perfect squares only end in 0, 1, 4, 5, 6 or 9, so a number ending in 2, 3, 7 or 8 cannot be a perfect square.
How do you find a square root quickly?
Use the last digit to narrow the possible units digit, use the leading digits to find the tens digit, then choose between the two candidates by comparing with the square of the number ending in 5.