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CUBES AND CUBE ROOTS 115

3
3
74088 = 2 × 2 × 2 × 3 × 3 × 3 × 7 × 7 × 7 = 2 × 3 × 7 = (2 × 3 × 7) 3
3
Therefore, 3 74088 = 2 × 3 × 7 = 42
Example 6: Find the cube root of 8000.
Solution: Prime factorisation of 8000 is 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5

So, 3 8000 = 2 × 2 × 5 = 20

Example 7: Find the cube root of 13824 by prime factorisation method.
Solution:
13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2 × 2 × 2 × 3 .
3
3
3
3
Therefore, 3 13824 = 2 × 2 × 2 × 3 = 24
THINK, DISCUSS AND WRITE

State true or false: for any integer m, m < m . Why?
3
2
7.3.2 Cube root of a cube number
If you know that the given number is a cube number then following method can be used.
Step 1 Take any cube number say 857375 and start making groups of three digits
starting from the right most digit of the number.

857 375
↓ ↓
second group first group

We can estimate the cube root of a given cube number through a step by
step process.
We get 375 and 857 as two groups of three digits each.
Step 2 First group, i.e., 375 will give you the one’s (or unit’s) digit of the required
cube root.
The number 375 ends with 5. We know that 5 comes at the unit’s place of a
number only when it’s cube root ends in 5.
So, we get 5 at the unit’s place of the cube root.
Step 3 Now take another group, i.e., 857.
We know that 9 = 729 and 10 = 1000. Also, 729 < 857 < 1000. We take
3
3
the one’s place, of the smaller number 729 as the ten’s place of the required
cube root. So, we get 3 857375 = 95 .

Example 8: Find the cube root of 17576 through estimation.

Solution: The given number is 17576.

Step 1 Form groups of three starting from the rightmost digit of 17576.