Square & Cube Roots
SSC GD Exam - Quantitative Aptitude
1. What is a Square?
When a number is multiplied by itself, the result is called its square.
Square of a = a × a = a²
2²
= 4
3²
= 9
4²
= 16
5²
= 25
2. What is a Square Root?
The square root of a number is a value that, when multiplied by itself, gives the original number.
√a = b if b × b = a
√9
= 3
√16
= 4
√25
= 5
3. Perfect Squares (1 to 30)
| Number | Square | Number | Square |
|---|---|---|---|
| 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 |
| 25 | 625 | 30 | 900 |
4. Shortcuts for Finding Square Roots
✅ Shortcut 1 – For Perfect Squares
If the number ends with 1, 4, 9, 6, or 5 → check which square ends with that digit.
✅ Shortcut 2 – Prime Factorization Method
• Write the number as a product of primes
• Pair the same numbers
• Multiply one from each pair
Example: √144 = √(2×2×2×2×3×3) = 2×2×3 = 12
✅ Shortcut 3 – Division Method
Used for large non-perfect numbers (like √2025). Divide into pairs and use long division.
5. What is a Cube?
When a number is multiplied by itself three times, it is called its cube.
Cube of a = a × a × a = a³
2³
= 8
3³
= 27
4³
= 64
5³
= 125
6. What is a Cube Root?
The cube root of a number is the value that, when multiplied three times by itself, gives the number.
∛a = b if b × b × b = a
∛8
= 2
∛27
= 3
∛64
= 4
7. Perfect Cubes (1 to 20)
| Number | Cube | Number | Cube |
|---|---|---|---|
| 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 |
8. Shortcut for Cube Roots
✅ Shortcut 1 – For Perfect Cubes (Last Digit Rule)
| Last Digit of Cube | Cube Root Ends With |
|---|---|
| 1 → | 1 |
| 4 → | 4 |
| 5 → | 5 |
| 6 → | 6 |
| 9 → | 9 |
| 0 → | 0 |
| 2 → | 8 |
| 3 → | 7 |
| 7 → | 3 |
| 8 → | 2 |
Example
Find ∛4096 → last digit 6 → root ends with 6, cube near 4096 is 16³
→ Answer = 16
9. Quick Recap
| Concept | Summary |
|---|---|
| Square | Number × itself |
| Square Root | Reverse of squaring |
| Cube | Number × itself × itself |
| Cube Root | Reverse of cubing |
| Perfect Squares | 1, 4, 9, 16, 25, ... |
| Perfect Cubes | 1, 8, 27, 64, 125, ... |
| Tip | Memorize squares (1–30) and cubes (1–20) for quick exams |
10. Practice Questions
Test your knowledge with these practice questions. Click on "View Answer" to check your understanding.
Q1. Find √81.
View Answer
9 ✅
Q2. Find √225.
View Answer
15 ✅
Q3. Find √625.
View Answer
25 ✅
Q4. Find √1444.
View Answer
38 ✅
Q5. Find ∛27.
View Answer
3 ✅
Q6. Find ∛512.
View Answer
8 ✅
Q7. Find the square of 18.
View Answer
18 × 18 = 324 ✅
Q8. Find the cube of 12.
View Answer
12 × 12 × 12 = 1728 ✅
Q9. Which of the following is a perfect
square?
(a) 42 (b) 49 (c) 51 (d) 53
View Answer
(b) 49 ✅
Q10. Which of the following is a
perfect cube?
(a) 64 (b) 72 (c) 90 (d) 125
View Answer
(a) 64 and (d) 125 ✅
Q11. Find the square root of 400.
View Answer
√400 = 20 ✅
Q12. Find ∛343.
View Answer
7 ✅
Q13. If 12² = 144, find (√144).
View Answer
12 ✅
Q14. The cube root of 1000 is?
View Answer
∛1000 = 10 ✅
Q15. Simplify √(49 × 9).
View Answer
√441 = 21 ✅
✅ Exam Strategy Tip
🟢 Memorize squares up to 30 and cubes up to 20 for instant recall
🟢 Use the last digit rule for quick cube root identification
🟢 Practice prime factorization for finding square roots of large numbers
🟢 Learn to recognize perfect squares and cubes by their ending digits
You've completed Square & Cube Roots Concepts!
Courage Tip: Mastering squares and cubes is fundamental for quick calculations in competitive exams. Regular practice with perfect squares (1-30) and cubes (1-20) will significantly improve your speed. Remember the patterns in ending digits and use prime factorization for complex roots. The more you practice, the faster you'll recognize these patterns during exams.
Master Square & Cube Roots for Competitive Exams!
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