Question:
Prove that $\frac{3}{\sqrt{5}}$ is irrational, given that $\sqrt{5}$ is irrational.
Solution:
Let us assume that $\frac{3}{\sqrt{5}}$ is a rational number.
Thus, $\frac{3}{\sqrt{5}}$ can be represented in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0, p$ and $q$ are co-prime numbers.
$\frac{3}{\sqrt{5}}=\frac{p}{q}$
$\Rightarrow p \sqrt{5}=3 q$
$\Rightarrow \sqrt{5}=\frac{3 q}{p}$
Since, $\frac{3 q}{p}$ is rational $\Rightarrow \sqrt{5}$ is rational
But, it is given that $\sqrt{5}$ is an irrational number.
Therefore, our assumption is wrong.
Hence, $\frac{3}{\sqrt{5}}$ is an irrational number.