Question

Prove that the following is irrational : $7\sqrt{5}$ [2 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark

We can prove $7\sqrt{5}$ is irrational by contradiction.

Lets suppose that $7\sqrt{5}$ is rational.

It means we have some co-prime integers a and b (b≠0) such that $7\sqrt{5}=\frac{a}{b}$

$\Rightarrow \sqrt{5}=\frac{a}{7b}$ ........ (1)

$\frac{a}{7b}$ is rational $\Rightarrow$ $\sqrt{5}$ is rational
But we know, $\sqrt{5}$ is irrational. It's a contradiction, which means our assumption is wrong.

$\therefore$ $7\sqrt{5}$ cannot be rational. Hence, it is irrational.