Question

Which of the following are not perfect cubes?
(i) $216$
(ii) $128$
(iii) $1000$
(iv) $100$
(v) $46656$

Answer 1

A perfect cube is defined as the product of three same integers. To check if a number $n$ is a perfect cube or not, we check whether an integer, when multiplied by itself thrice, gives the number $n$ or not.

(i) $216=2\times 2\times 2\times 3\times 3\times 3$

$=2^3\times 3^3$
$={(2\times 3)}^3$ [Since $a^m\times b^m=(ab{)}^m$]
So, $216$ is a perfect cube.
(ii) $128=2\times 2\times 2\times 2\times 2\times 2\times 2$
$=2^3\times 2^3\times 2$
$=4^3\times 2$
So, $128$ is not a perfect cube.

(iii) $1000=2\times 2\times 2\times 5\times 5\times 5$
$=2^3\times 5^3$
$=(2\times 5{)}^3$ [since, $a^m\times b^m=(ab{)}^m$
$={10}^3$
So, $1000$ is a perfect cube.
(iv) $100=2\times 2\times 5\times 5={10}^2$
So, $100$ is not a perfect cube
(v) $46656=2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\times 3\times 3\times 3$
$=2^6\times 3^6$
$={\left(2^2\right)}^3\times {\left(3^2\right)}^3$
$=4^3\times 9^3$ [Since, $(a^m{)}^n=a^{m\times n}$]
$=(9\times 4{)}^3$ [ Since, $a^m\times b^m=(ab{)}^m$]
$=(36{)}^3$
So, $46656$ is a perfect cube.