A perfect cube is an integer whose cube root is an integer. For example, 27, 64 and 125 are perfect cubes. If p and q are perfect cubes, which of the following will not necessarily be a perfect cube?

Question: A perfect cube is an integer whose cube root is an integer. For example, 27, 64 and 125 are perfect cubes. If p and q are perfect cubes, which of the following will not necessarily be a perfect cube?

Solution:
Given,
p and q are perfect cubes.

So, let,
p = a³ and q = b³, where a and b are integers.

Now check each option:

ক) 8p
= 2³ × a³
= (2a)³
So, 8p is always a perfect cube.

খ) pq
= a³ × b³
= (ab)³
So, pq is always a perfect cube.

গ) pq + 27
Here, pq is a perfect cube and 27 is also a perfect cube.
But the sum of two perfect cubes is not always a perfect cube.

For example,
if p = 1 and q = 1,
then pq + 27 = 1 × 1 + 27
= 28
28 is not a perfect cube.

ঘ) - p
= - a³
= (- a)³
So, - p is always a perfect cube.

Therefore, pq + 27 will not necessarily be a perfect cube.

Correct Answer: গ) pq + 27