Question

How do you simplify the radical expression: $\dfrac{\sqrt{120}}{\sqrt{5}}$?

Answer 1

Hint: In this problem we need to simplify the given radical expression. We can observe that the given expression contains a fraction of two root values. So, we will consider both numerator and denominator individually. After that we will factorize the value $120$ which is in the numerator. After we will write the value $120$ as a product of two numbers based on the value in the denominator in order to simplify the fraction. Now we will apply the exponential formula $\sqrt{a.b}=\sqrt{a}.\sqrt{b}$ and simplify the equation.

Complete step by step solution:
Given radical expression is $\dfrac{\sqrt{120}}{\sqrt{5}}$.
In the above fraction we have $\sqrt{120}$ as numerator and $\sqrt{5}$ as denominator.
Considering the numerator which is $\sqrt{120}$.
We can write the value $120$ from its factors as
$\Rightarrow 120={{2}^{3}}\times 3\times 5$
In the denominator we have the value $\sqrt{5}$. So, we are going to write the value $120$ as
$\begin{align}
  & \Rightarrow 120=\left( {{2}^{3}}\times 3 \right)\times 5 \\
 & \Rightarrow 120=\left( 8\times 3 \right)\times 5 \\
 & \Rightarrow 120=24\times 5 \\
\end{align}$
Now the value of $\sqrt{120}$ will be calculated by applying square root on both sides of the above equation, then we will get
$\Rightarrow \sqrt{120}=\sqrt{24\times 5}$
We have the exponential formula $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$. Applying this formula in the above equation, then we will get
$\Rightarrow \sqrt{120}=\sqrt{24}\times \sqrt{5}$
Now the value of given radical expression can be calculated by dividing the above equation with $\sqrt{5}$ on both sides of the above equation, then we will have
$\Rightarrow \dfrac{\sqrt{120}}{\sqrt{5}}=\dfrac{\sqrt{24}\times \sqrt{5}}{\sqrt{5}}$
Cancelling the value $\sqrt{5}$ which is in both numerator and denominator, then we will get
$\Rightarrow \dfrac{\sqrt{120}}{\sqrt{5}}=\sqrt{24}$
We can write $24=4\times 6$ in the above equation, then we will have
$\begin{align}
  & \Rightarrow \dfrac{\sqrt{120}}{\sqrt{5}}=\sqrt{4\times 6} \\
 & \Rightarrow \dfrac{\sqrt{120}}{\sqrt{5}}=2\sqrt{6} \\
\end{align}$
Hence the simplified form of the given radical expression $\dfrac{\sqrt{120}}{\sqrt{5}}$ is $2\sqrt{6}$.

Note: We can also simplify the given value by applying the formula $\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}$. After applying this formula, we will calculate the value of $\dfrac{120}{5}$ by dividing $120$ with $5$, then we will simplify the equation to get the required result.