Which of the following numbers are cubes of negative integers
(i) −64
(ii) −1056
(iii) −2197
(iv) −2744
(v) −42875
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,
(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:
$64=\{2 \times 2 \times 2\} \times\{2 \times 2 \times 2\}$
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
Now, collect one factor from each triplet and multiply, we ge
$2 \times 2=4$
This implies that 64 is a cube of 4.
Thus,
(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:
$1056=\{2 \times 2 \times 2\} \times 2 \times 2 \times 3 \times 11$
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
(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:
$2197=\{13 \times 13 \times 13\}$
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
Now, collect one factor from each triplet and multiply, we get 13.
This implies that 2197 is a cube of 13.
Thus,
(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:
$2744=\{2 \times 2 \times 2\} \times\{7 \times 7 \times 7\}$
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
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,
(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:
$42875=\{5 \times 5 \times 5\} \times\{7 \times 7 \times 7\}$
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
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,