Question

Which of the following numbers are cubes of negative integers
(i) −64
(ii) −1056
(iii) −2197
(iv) −2744
(v) −42875

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.

(i)
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.

(ii)
On factorising 1056 into prime factors, we get:
$1056=2\times 2\times 2\times 2\times 2\times 3\times 11$
On grouping the factors in triples of equal factors, we get:​
$1056=\left\{2\times 2\times 2\right\}\times 2\times 2\times 3\times 11$
It is evident that the prime factors of 1056 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 1056 is not a perfect cube. This implies that $-$1056 is not a perfect cube as well.

(iii)
On factorising 2197 into prime factors, we get:
$2197=13\times 13\times 13$
On grouping the factors in triples of equal factors, we get:​
$2197=\left\{13\times 13\times 13\right\}$
It is evident that the prime factors of 2197 can be grouped into triples of equal factors and no factor is left over. Therefore, 2197 is a perfect cube. This implies that $-$2197 is also a perfect cube.

Now, collect one factor from each triplet and multiply, we get 13.
This implies that 2197 is a cube of 13.
Thus, $-$2197 is the cube of $-$13.

(iv)
On factorising 2744 into prime factors, we get:
$2744=2\times 2\times 2\times 7\times 7\times 7$
On grouping the factors in triples of equal factors, we get:​
$2744=\left\{2\times 2\times 2\right\}\times \left\{7\times 7\times 7\right\}$
It is evident that the prime factors of 2744 can be grouped into triples of equal factors and no factor is left over. Therefore, 2744 is a perfect cube. This implies that $-$2744 is also a perfect cube.

Now, collect one factor from each triplet and multiply, we get:
$2\times 7=14$
This implies that 2744 is a cube of 14.
Thus, $-$2744 is the cube of $-$14.

(v)
On factorising 42875 into prime factors, we get:
$42875=5\times 5\times 5\times 7\times 7\times 7$
On grouping the factors in triples of equal factors, we get:​
$42875=\left\{5\times 5\times 5\right\}\times \left\{7\times 7\times 7\right\}$
It is evident that the prime factors of 42875 can be grouped into triples of equal factors and no factor is left over. Therefore, 42875 is a perfect cube. This implies that $-$42875 is also a perfect cube.

Now, collect one factor from each triplet and multiply, we get:
$5\times 7=35$
This implies that 42875 is a cube of 35.
Thus, $-$42875 is the cube of $-$35.