Question

Rationalize the denominator of the following and write the answer:
${\displaystyle \frac{3\sqrt{2}}{\sqrt{5}}}$

Answer 1

Hint:To rationalize the denominator means we have to make the denominator an integer by multiplying some numbers both in numerator and denominator. The number can be a single irrational number or some addition or subtraction operation performed on an irrational number.

Complete step-by-step answer:
Let’s start to solve this question by analysing what number is needed to rationalize the denominator.
Let’s take an example just to understand it better.
If there is an number in the form $\begin{array}{rl}& {\displaystyle \frac{\sqrt{a}\pm \sqrt{b}}{\sqrt{c}+\sqrt{d}}}\\ & \end{array}$ then we have to multiply by a number which can make the denominator into an integer.
In this case if we multiply by $(\sqrt{c}-\sqrt{d})$ then the denominator becomes $(\sqrt{c}+\sqrt{d})(\sqrt{c}-\sqrt{d})$ then by using identity $(a+b)(a-b)$=$a^2-b^2$ which becomes $c-d$ and this is an integer.
So, this is how rationalization is done. But the question which is given is easier to solve than the above given example.
Now, in this question we have given ${\displaystyle \frac{3\sqrt{2}}{\sqrt{5}}}$, to rationalize this we have to multiply the numerator and denominator by $\sqrt{5}$ to make it into an integer.
After multiplying we get,
$\begin{array}{rl}& \Rightarrow \left({\displaystyle \frac{3\sqrt{2}}{\sqrt{5}}}\right).{\displaystyle \frac{\sqrt{5}}{\sqrt{5}}}\\ & \Rightarrow {\displaystyle \frac{3\sqrt{10}}{\sqrt{5}.\sqrt{5}}}\\ & \Rightarrow {\displaystyle \frac{3\sqrt{10}}{5}}\end{array}$
Hence, our final answer after rationalizing the denominator is ${\displaystyle \frac{3\sqrt{10}}{5}}$ .

Note: The question can also be solved by multiplying some different numbers just for a better understanding of the concept. Like we can multiply $n\sqrt{5}$ where n is any natural number, try to solve this question by putting different values of n to get new numbers and check whether this is true or not.