Question:
Prove that $\frac{2 \sqrt{3}}{5}$ is irrational.
Solution:
Here we have to prove that the number $\frac{2 \sqrt{3}}{5}$ is an irrational number.
Now let us suppose that $\frac{2 \sqrt{3}}{5}=x$, where $x$ is a rational number, then
$2 \sqrt{3}=5 x$
$\Rightarrow \sqrt{3}=\frac{5 x}{2}$ ..........(1)
As $x$ is rational number, therefore form equation (1), so is $\frac{5 x}{2}$ and $\sqrt{3}$ is also a rational number which is a contradiction as $\sqrt{3}$ is an irrational number.
Therefore $\frac{2 \sqrt{3}}{5}$ is an irrational number.