Question

Show that the following integer is cube of negative integer. Also, find the integer whose cube is the given integer −2744000 .

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 2744000 into prime factors, we get:

\[2744000 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 7 \times 7 \times 7\] 

On grouping the factors in triples of equal factors, we get:​

\[2744000 = \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 5 \times 5 \times 5 \right\} \times \left\{ 7 \times 7 \times 7 \right\}\]

It is evident that the prime factors of 2744000 can be grouped into triples of equal factors and no factor is left over. Therefore, 2744000 is a perfect cube. This implies that - 2744000 is also a perfect cube. 

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

\[2 \times 2 \times 5 \times 7 = 140\]

This implies that 2744000 is a cube of 140.

Thus, - 2744000 is the cube of  - 140.