Question

Which of the following represents the power of product rule?
A. ${{\left( x\times y \right)}^{a}}={{x}^{a}}\times y$
B. ${{\left( x\times y \right)}^{a}}=x\times {{y}^{a}}$
C. ${{\left( x\times y \right)}^{a}}={{x}^{a}}+{{y}^{a}}$
D. ${{\left( x\times y \right)}^{a}}={{x}^{a}}\times {{y}^{a}}$

Answer 1

Hint: In this problem we need to check for the equation which represents the power of product rule. This rule is used to calculate the power of a value which is obtained as a result of multiplying two variables. So we will first assume any two variables and calculate the product of the variables. Now calculate any power of the obtained result. Now consider the given option and calculate the values according to the given expression. Now check both the values to get the required result.

Complete step by step answer:
Given that, power of product rules.
This rule is used to calculate the power of a value which is obtained as a result of multiplying two variables.
Let us assume the variables $x=2$ , $y=3$ and $a=2$ .
Now the value of ${{\left( x\times y \right)}^{a}}$ is calculated by
${{\left( x\times y \right)}^{a}}={{\left( 2\times 3 \right)}^{2}}$
Simplifying the above equation, then we will have
$\begin{align}
  & {{\left( x\times y \right)}^{a}}={{6}^{2}} \\
 & \Rightarrow {{\left( x\times y \right)}^{a}}=36 \\
\end{align}$
Now consider the first option which says ${{\left( x\times y \right)}^{a}}={{x}^{a}}\times y$.
The value of ${{x}^{a}}\times y$ is given by
${{x}^{a}}\times y={{2}^{2}}\times 3$
Simplifying the above equation, then we will get
$\begin{align}
  & {{x}^{a}}\times y=4\times 3 \\
 & \Rightarrow {{x}^{a}}\times y=12 \\
\end{align}$
Now consider the second option which says ${{\left( x\times y \right)}^{a}}=x\times {{y}^{a}}$.
The value of $x\times {{y}^{a}}$ is given by
$x\times {{y}^{a}}=2\times {{3}^{2}}$
Simplifying the above equation, then we will have
$\begin{align}
  & x\times {{y}^{a}}=2\times 9 \\
 & \Rightarrow x\times {{y}^{a}}=18 \\
\end{align}$
Now consider the third option which says ${{\left( x\times y \right)}^{a}}={{x}^{a}}+{{y}^{a}}$
The value of ${{x}^{a}}+{{y}^{a}}$ is given by
${{x}^{a}}+{{y}^{a}}={{2}^{2}}+{{3}^{2}}$
Simplifying the above equation, then we will get
$\begin{align}
  & {{x}^{a}}+{{y}^{a}}=4+9 \\
 & \Rightarrow {{x}^{a}}+{{y}^{a}}=13 \\
\end{align}$
Considering the fourth option which says ${{\left( x\times y \right)}^{a}}={{x}^{a}}\times {{y}^{a}}$
The value of ${{x}^{a}}\times {{y}^{a}}$ is given by
${{x}^{a}}\times {{y}^{a}}={{2}^{2}}\times {{3}^{2}}$
Simplifying the above equation, then we will have
$\begin{align}
  & {{x}^{a}}\times {{y}^{a}}=4\times 9 \\
 & \Rightarrow {{x}^{a}}\times {{y}^{a}}=36 \\
\end{align}$
On observing the values given by the fourth options we can say that the value of ${{\left( x\times y \right)}^{a}}$ is equal to ${{x}^{a}}\times {{y}^{a}}$.

Hence the power of product rules is represented by ${{\left( x\times y \right)}^{a}}={{x}^{a}}\times {{y}^{a}}$.

So, the correct answer is “Option D”.

Note: From the statement of the power of product rule we can simply identify the correct answer. The power of product rule says that “if the product of the bases is powered by the same exponent, then the result is multiplication of all the bases, each powered by the given exponent”. From this statement also we can say that the power of product rules is represented by ${{\left( x\times y \right)}^{a}}={{x}^{a}}\times {{y}^{a}}$.