Question

How do you simplify the radical expression by rationalizing the denominator $\dfrac{2}{\sqrt{30}}$?

Answer 1

Hint: To simplify the given fraction $\dfrac{2}{\sqrt{30}}$ by rationalizing the denominator. Now, to solve this problem, we need to rationalize the given fraction, rationalization means we have to multiply the numerator and denominator by $\sqrt{30}$. And hence, solve the multiplication to simplify it further.

Complete step by step solution:
The fraction given in the above problem is as follows:
$\dfrac{2}{\sqrt{30}}$
Simplifying the above expression by rationalizing the denominator and rationalization of the denominator is done by multiplying and dividing the denominator of the given fraction. So, multiplying $\sqrt{30}$ in the numerator and the denominator in the above fraction we get,
$\Rightarrow \dfrac{2}{\sqrt{30}}\times \dfrac{\sqrt{30}}{\sqrt{30}}$ ………… (1)
We know that when we multiply two square root terms then we get the term itself. The statement which we have just said looks in the mathematical form as follows:
$\Rightarrow \sqrt{a}\times \sqrt{a}=a$
Applying the above relation in the denominator of the relation (1) we get,
$\Rightarrow \dfrac{2\sqrt{30}}{30}$
If you look carefully at the above expression then you will find that the numerator and denominator will be divided by 2. The numerator is 2 itself so when divided by 2 will give 1 as the answer and when we divide 30 by 2 then we will get 15 so using these division by 2 results in the above expression we get,
$\Rightarrow \dfrac{\sqrt{30}}{15}$
Hence, we have simplified the given expression to $\dfrac{\sqrt{30}}{15}$.

Note: If in the above problem instead of $\sqrt{30}$, $2+\sqrt{3}$ is given then the way we are going to rationalize the denominator by multiplying and dividing by the conjugate of $2+\sqrt{3}$ and we know that the conjugate of $2+\sqrt{3}$ is $2-\sqrt{3}$ so rationalizing will get us:
$=\dfrac{2}{2+\sqrt{3}}\times \dfrac{2-\sqrt{3}}{2-\sqrt{3}}$
In the denominator, we can apply the following algebraic identity as follows:
$\Rightarrow \left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$
Then the denominator of the above expression will get simplified to:
\[\begin{align}
  & =\dfrac{2\left( 2-\sqrt{3} \right)}{{{2}^{2}}-{{\left( \sqrt{3} \right)}^{2}}} \\
 & =\dfrac{4-2\sqrt{3}}{4-3} \\
 & =\dfrac{4-2\sqrt{3}}{1} \\
\end{align}\]