Question

Which of the following number is cube of negative integer - 64 .

Answer 1

In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer m, -m³ is the cube of - m.

On factorising 64 into prime factors, we get: \[64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2\] On grouping the factors in triples of equal factors, we get: \[64 = \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 2 \times 2 \times 2 \right\}\] It is evident that the prime factors of 64 can be grouped into triples of equal factors and no factor is left over. Therefore, 64 is a perfect cube. This implies that - 64 is also a perfect cube.

Now, collect one factor from each triplet and multiply, we get: \[2 \times 2 = 4\]  This implies that 64 is a cube of 4.
Thus, -64 is the cube of - 4.