Quick Concept Summary
Square roots and cube roots appear in simplification, Pythagorean theorem problems, and mensuration. Memorising perfect squares and cubes saves time in exams.
Square root (√): √n = x means x × x = n
Cube root (∛): ∛n = x means x × x × x = n
Perfect Squares — Memorise These
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 11 | 121 |
| 2 | 4 | 12 | 144 |
| 3 | 9 | 13 | 169 |
| 4 | 16 | 14 | 196 |
| 5 | 25 | 15 | 225 |
| 6 | 36 | 16 | 256 |
| 7 | 49 | 17 | 289 |
| 8 | 64 | 18 | 324 |
| 9 | 81 | 19 | 361 |
| 10 | 100 | 20 | 400 |
Extended: 21²=441, 22²=484, 23²=529, 24²=576, 25²=625
Perfect Cubes — Memorise These
| n | n³ |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
| 11 | 1331 |
| 12 | 1728 |
| 15 | 3375 |
Key Formulas
| Formula | Example |
|---|---|
| √(a×b) = √a × √b | √(4×9) = 2×3 = 6 |
| √(a/b) = √a / √b | √(25/4) = 5/2 |
| (√a)² = a | (√7)² = 7 |
| √a² = a (a > 0) | √49 = 7 |
| ∛(a×b) = ∛a × ∛b | ∛(8×27) = 2×3 = 6 |
Worked Examples
Example 1 — Square Root by Prime Factorisation
Q: Find √1764.
1764 = 2 × 882 = 2 × 2 × 441 = 4 × 441 = 4 × 9 × 49
1764 = 2² × 3² × 7²
√1764 = 2 × 3 × 7 = 42
Example 2 — Cube Root by Prime Factorisation
Q: Find ∛13824.
13824 = 2 × 6912 = 2² × 3456 = 2³ × 1728 = 2³ × 12³
Wait, let's factor step by step: 13824 ÷ 2 = 6912 ÷ 2 = 3456 ÷ 2 = 1728 ÷ 2 = 864 ÷ 2 = 432 ÷ 2 = 216 ÷ 2 = 108...
Actually: 13824 = 8 × 1728 = 2³ × 12³ = (2×12)³ = 24³
∛13824 = 24
Example 3 — Simplification Using Square Root Properties
Q: Simplify: √(75) + √(48) − √(27)
√75 = √(25×3) = 5√3
√48 = √(16×3) = 4√3
√27 = √(9×3) = 3√3
Result = 5√3 + 4√3 − 3√3 = 6√3
Example 4 — Finding Missing Value
Q: If √(2025) = 45, find √(20.25).
20.25 = 2025/100
√(20.25) = √2025 / √100 = 45/10 = 4.5
Example 5 — Perfect Square Check
Q: Is 2916 a perfect square? If yes, find its square root.
Unit digit of 2916 is 6 → possible perfect square (ends in 6).
2916 = 4 × 729 = 4 × 729 = 2² × 27² = (2×27)² = 54²
√2916 = 54 ✓
Example 6 — Approximation
Q: The square root of 150 lies between which two consecutive integers?
12² = 144, 13² = 169
Since 144 < 150 < 169, √150 lies between 12 and 13 (closer to 12).
Shortcuts and Tricks
Trick 1 — Unit digit of perfect squares
Perfect squares always end in: 0, 1, 4, 5, 6, 9
If a number ends in 2, 3, 7, or 8 → NOT a perfect square (rule it out instantly).
Trick 2 — Unit digit of square root
| Square ends in | Square root ends in |
|---|---|
| 1 | 1 or 9 |
| 4 | 2 or 8 |
| 5 | 5 |
| 6 | 4 or 6 |
| 9 | 3 or 7 |
| 0 | 0 |
Trick 3 — Finding √ of numbers near perfect squares
√(n² ± small x) ≈ n ± x/(2n)
Example: √(101) ≈ 10 + 1/20 = 10.05
Trick 4 — Cube root unit digit trick
| Cube ends in | Cube root ends in |
|---|---|
| 1 | 1 |
| 8 | 2 |
| 7 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 3 | 7 |
| 2 | 8 |
| 9 | 9 |
| 0 | 0 |
Cube root unit digit has a unique mapping — memorise this for quick answers.
TNPSC / Placement Exam Pattern
TNPSC: 1–2 questions from simplification sections. Often simplify √(large number) or check if a number is a perfect square/cube. Sometimes combined with fractions.
SSC/Bank: Long division method may be needed for imperfect squares. Finding approximate values between two integers is also common.
Practice Questions
| # | Question | Answer |
|---|---|---|
| 1 | √(2304) = ? | 48 |
| 2 | ∛(9261) = ? | 21 |
| 3 | √(0.0016) = ? | 0.04 |
| 4 | Simplify: √(32) + √(8) | 6√2 |
| 5 | Is 3136 a perfect square? If yes, find √3136. | Yes, 56 |
| 6 | ∛(0.000343) = ? | 0.07 |
| 7 | √(576) × ∛(512) = ? | 24 × 8 = 192 |
| 8 | Between which integers does √200 lie? | 14 and 15 |
| 9 | Simplify: √(45) − √(20) + √(80) | 5√5 |
| 10 | What is the least number to be added to 1300 to make it a perfect square? | 36 (since 36²=1296, 37²=1369; add 69) |