1. What is a Square?
When a number is multiplied by itself, the result is called the square of that number.
Here, 25 is the square of 5.
2. Important Squares
| 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 |
3. What is a Square Root?
The square root of a number is the number which, when multiplied by itself, gives the original number.
This is because:
Therefore: √25 = 5
4. Perfect Square
A number whose square root is a whole number is called a perfect square.
For example, √100 = 10. Therefore, 100 is a perfect square.
5. Important Properties
Property 2: The square of an odd number is odd.
Property 3: A perfect square cannot end in 2, 3, 7 or 8.
Property 4: The square of a number ending in 5 always ends in 25.
25² = 625
35² = 1225
6. Square Root by Prime Factorisation
To find the square root using prime factorisation, first find the prime factors of the number. Then make pairs of identical factors. Take one factor from each pair outside the square-root sign.
144 = (2 × 2) × (2 × 2) × (3 × 3)
√144 = 2 × 2 × 3
√144 = 12
√225 = 3 × 5
√225 = 15
7. Square Root by Long Division
The long-division method is useful for large numbers and for numbers that are not easily factorised.
5 | 29
The largest square less than 5 is:
2² = 4
Subtract:
5 − 4 = 1
Bring down 29:
129
Double 2:
2 × 2 = 4
Try 3:
43 × 3 = 129
Therefore: √529 = 23
8. Estimating a Square Root
If a number is not a perfect square, we can find an approximate value by comparing it with nearby perfect squares.
8² = 64
Therefore: 7 < √50 < 8
Since 50 is very close to 49:
√50 ≈ 7.07
9. Square Root Rules
= √25 × √4
= 5 × 2
= 10
10. What is a Cube?
When a number is multiplied by itself three times, the result is called its cube.
Therefore, 3³ = 27
11. Important Cubes
| 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 |
12. What is a Cube Root?
The cube root of a number is the number which, when multiplied by itself three times, gives the original number.
This is because:
Therefore: ∛27 = 3
13. Perfect Cube
A number whose cube root is a whole number is called a perfect cube.
14. Cube Root by Prime Factorisation
To find a cube root using prime factorisation, make groups of three identical prime factors. Take one factor from each group.
216 = (2 × 2 × 2) × (3 × 3 × 3)
∛216 = 2 × 3
∛216 = 6
∛1000 = 2 × 5
∛1000 = 10
15. Cube Root of a Negative Number
Cube roots can be taken for negative numbers because the product of three negative numbers is negative.
Therefore: ∛(-27) = -3
Similarly: ∛(-125) = -5
16. Cube Root Using the Last Digit
The last digit of a perfect cube can help us identify the last digit of its cube root.
| Last Digit of Cube | Last Digit of Cube Root |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
| 3 | 7 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 3 |
| 8 | 2 |
| 9 | 9 |
2 ↔ 8
3 ↔ 7
The other digits remain the same.
12 | 167
2³ = 8 and 3³ = 27. Therefore, the first digit is 2.
The number ends in 7. A cube ending in 7 has a cube root ending in 3.
Therefore: ∛12167 = 23
Check: 23 × 23 × 23 = 12167
17. Square Root vs Cube Root
| Square Root | Cube Root |
|---|---|
| Symbol: √ | Symbol: ∛ |
| Number multiplied twice | Number multiplied three times |
| √49 = 7 | ∛343 = 7 |
| Factors grouped in pairs | Factors grouped in threes |
| Negative number has no real square root | Negative number has a real cube root |
18. Important Formulas
√(a²) = a
a³ = a × a × a
∛(a³) = a
19. Practice Questions
1. Find 12².
2. Find 18².
3. Find 25².
4. Find 32².
5. Find 45².
6. Find √81.
7. Find √144.
8. Find √225.
9. Find √400.
10. Find √625.
11. Find √900.
12. Find √1225.
13. Find √1600.
14. Find √2025.
15. Find √2500.
16. Find 4³.
17. Find 6³.
18. Find 8³.
19. Find 12³.
20. Find 15³.
21. Find ∛27.
22. Find ∛125.
23. Find ∛216.
24. Find ∛512.
25. Find ∛729.
26. Find ∛1000.
27. Find ∛1728.
28. Find ∛2197.
29. Find ∛3375.
30. Find ∛8000.
20. Quick Revision
Square Root: The number which gives the original number when multiplied by itself.
Cube: A number multiplied by itself three times.
Cube Root: The number which gives the original number when multiplied by itself three times.
Remember: √25 = 5 because 5² = 25
∛125 = 5 because 5³ = 125
For square roots, group prime factors in pairs.
For cube roots, group prime factors in groups of three.
6. 9 7. 12 8. 15 9. 20 10. 25
11. 30 12. 35 13. 40 14. 45 15. 50
16. 64 17. 216 18. 512 19. 1728 20. 3375
21. 3 22. 5 23. 6 24. 8 25. 9
26. 10 27. 12 28. 13 29. 15 30. 20
