Question

How do you simplify $\dfrac{2{{r}^{4}}}{4{{r}^{2}}}$?

Answer 1

Hint: The given expression is a fraction in terms of the variable $r$. So firstly, its domain needs to be obtained by putting the denominator of the given expression not equal to zero. Then, for simplifying it, we have to use the negative exponent rule, which is given as $\dfrac{1}{{{a}^{m}}}={{a}^{-m}}$, and take the ${{r}^{2}}$ term in the numerator to the denominator. Then, we have to use the second property of the exponents given by ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$ to finally simplify the given expression.

Complete step by step solution:
Let us write the given expression as
$f\left( r \right)=\dfrac{2{{r}^{4}}}{4{{r}^{2}}}$
We know that the denominator can never be equal to zero. So from the above expression we can write
$\Rightarrow 4{{r}^{2}}\ne 0$
Dividing both the sides by $4$ we get
$\begin{align}
  & \Rightarrow \dfrac{4{{r}^{2}}}{4}\ne \dfrac{0}{4} \\
 & \Rightarrow {{r}^{2}}\ne 0 \\
 & \Rightarrow r\ne 0 \\
\end{align}$
Therefore, the domain of the given expression is $r\in R-\left\{ 0 \right\}$.
Now, we can simplify the given expression by considering
$\Rightarrow f\left( r \right)=\dfrac{2{{r}^{4}}}{4{{r}^{2}}}$
Cancelling $2$ in the numerator by $4$ in the denominator, we get
$\Rightarrow f\left( r \right)=\dfrac{{{r}^{4}}}{2{{r}^{2}}}$
From the negative exponent rule $\dfrac{1}{{{a}^{m}}}={{a}^{-m}}$, we can write the above expression as
\[\Rightarrow f\left( r \right)=\dfrac{{{r}^{4}}{{r}^{-2}}}{2}\]
Now, we know from the common base rule that ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$. Therefore, the above expression can be written as
\[\begin{align}
  & \Rightarrow f\left( r \right)=\dfrac{{{r}^{4-2}}}{2} \\
 & \Rightarrow f\left( r \right)=\dfrac{{{r}^{2}}}{2} \\
\end{align}\]

Hence, the given expression is simplified as \[\dfrac{{{r}^{2}}}{2}\], with the condition $r\in R-\left\{ 0 \right\}$

Note: Instead of using the two rules of exponents, used in the above solution, we can also expand the numerator and denominator according to the exponent on $r$. Then, on cancelling out the terms common to the numerator and the denominator, we will get the same simplified expression as obtained in the above solution. Also, do not forget to determine the domain of the fractional expression given in the question.