Question

Show that the following integer is cube of negative integer. Also, find the integer whose cube is the given integer −5832 .

 

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 5832 into prime factors, we get:

\[5832 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\] On grouping the factors in triples of equal factors, we get:

\[5832 = \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 3 \times 3 \times 3 \right\} \times \left\{ 3 \times 3 \times 3 \right\}\] It is evident that the prime factors of 5832 can be grouped into triples of equal factors and no factor is left over. Therefore, 5832 is a perfect cube. This implies that - 5832 is also a perfect cube. 

Now, collect one factor from each triplet and multiply, we get: 

 \[2 \times 3 \times 3 = 18\] 

This implies that 5832 is a cube of 18.
Thus, - 5832 is the cube of - 18.