Question

Prove that $\frac{2}{\sqrt{7}}$ is irrational.

Answer 1

Let us assume that $\frac{2}{\sqrt{7}}$ is a rational number.

$\frac{2}{\sqrt{7}}$= $\frac{2}{\sqrt{7}}\times \frac{\sqrt{7}}{\sqrt{7}}=\frac{2\sqrt{7}}{7}$( Rationalising)

Now,$\frac{2\sqrt{7}}{7}=\frac{p}{q}$ ( as rational no. can be written in the form of $\frac{p}{q}$)

$2\sqrt{7}=\frac{7p}{q}$
$\sqrt{7}=\frac{7p}{2q}$

Here LHS is irrational but RHS is rational.

This contradicts the statement.
Our assumption is wrong.

$\frac{2}{\sqrt{7}}$ is an irrational number.