Question

Show that $3\sqrt{7}$ is an irrational number

Answer 1

Let us assume that $3\sqrt{7}$ is rational.

$3\sqrt{7}=\frac{a}{b}$

Rearranging, we get $\sqrt{7}=\frac{a}{3b}$

Since 3, a and b are integers, $\frac{a}{3b}$ can be written in the form of $\frac{p}{q}$, so $\frac{a}{3b}$ is rational, and so $\sqrt{7}$ is rational.

But this contradicts that $\sqrt{7}$ is irrational. So, we conclude that $3\sqrt{7}$ is irrational.