Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube. (i) 81 (ii) 128 (iii) 135 (iv) 192 (v) 704

(i) We have 81 = 3 x 3 x 3 x 3

Grouping the prime factors of 81 into triples, we are left with 3.

∴ 81 is not a perfect cube
Now, [81] $divided by$ 3= [3 x 3 x 3 x 3] $divided by$ 3
or 27 = 3 x 3 x 3
i.e. 27 is a perfect cube
Thus, the required smallest number is 3

(ii) we have 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2

Grouping the prime factors of 128 into triples, we are left with 2
∴ 128 is not a perfect cube
Now, [128] $divided by$ 2 = [2 x 2 x 2 x 2 x 2 x 2 x 2] $divided by$ 2
or 64 = 2 x 2 x 2 x 2 x 2 x 2
i.e. 64 is a perfect cube
∴ the smallest required number is 2.

(iii) we have 135 = 3 x 3 x 3 x 5

Grouping the prime factors of 135 into triples, we are left over with 5.
∴ 135 is not a perfect cube
Now, [135] $divided by$ 5 = [ 3 x 3 x 3 x 5] $divided by$ 5
or 27 = 3 x 3 x 3
i.e. 27 is a perfect cube.
Thus, the required smallest number is 5
(iv) We have 192 = 2 x 2 x 2 x 2 x 2 x 2 x 3

Grouping the prime factors of 192 into triples, 3 is left over.
∴ 192 is not a perfect cube.
Now, [192] $divided by$ 3= [2 x 2 x 2 x 2 x 2 x 2 x 3] $divided by$ 3
or 64 = 2 x 2 x 2 x 2 x 2 x 2
i.e. 64 is a perfect cube.
Thus, the required smallest number is 3.
(v) We have 704 = 2 x 2 x 2 x 2 x 2 x 2 x 11

Grouping the prime factors of 704 into triples, 11 is left over
∴ [704] $divided by$ 11 = [2 x 2 x 2 x 2 x 2 x 2 x 11] $divided by$ 11
or 64 = 2 x 2 x 2 x 2 x 2 x 2
i.e. 64 is a perfect cube
Thus, the required smallest number is 11.