Question

Show that $4\sqrt{2}$ is an irrational number.

Answer 1

Assume that,

$4\sqrt{2}$ is a rational number.

Then, there exists coprime positive integers p & q such that

$4\sqrt{2}=\frac{p}{q}$

$\sqrt{2}=\frac{p}{4q}$ ($\because$ p & q are integers)

$\Rightarrow \frac{p}{4q}$ is rational

$\Rightarrow \sqrt{2}$ is rational

This contradict the fact that $\sqrt{2}$

is irrational. so our assumption is incorrect.

Hence $4\sqrt{2}$ is irritational.