Question

A product of two numbers is given as $x^4{y}^2$, such that $x$ and $y$ are positive integers. Which of the following should be multiplied with the given product to make it a perfect cube?

• A. $x^6{y}^3$
• B. $xy$
• C. $x^{2}y$
• D. $y$

Answer 1

The correct option is C $x^{2}y$

Any number is said to be a perfect cube if each of its factors repeat three times, i.e., if $a$ and $b$ are two factors of any positive integer $z$, then $z$ will be a perfect cube if it can be written as,

$z=a^3{b}^3\text{or}{a}^6{b}^6\text{or}{a}^9{b}^9$ or so on.

Consider the given product $x^4{y}^2$. It can be represented as,

$x^4{y}^2=x\times x\times x\times x\times y\times y$

This can be a perfect cube if each of its factors must be repeated thrice or as a multiple of $3$, i.e., the factors must repeat $3,6,9$ or so on times.

$x^4$ must be multiplied with $x^2$, so that it can become a perfect cube and $y^2$ must be multiplied by $y$, i.e.,

$\begin{array}{cc}\left(x^4{y}^2\right)\left(x^{2}y\right)& =(x\times x\times x\times x\times y\times y)(x\times x\times y)\\ & =x\times x\times x\times x\times x\times x\times y\times y\times y\\ & =x^6{y}^3\end{array}$

Thus, $x^4{y}^2$ must be multiplied by $x^{2}y$ to make it a perfect cube which is $x^6{y}^3$