Question

Prove that the following is irrational:

${\displaystyle 7\sqrt{5}}$

Answer 1

Let a = ${\displaystyle 7\sqrt{5}}$ be a rational number.

We can find two co-prime integers a and b such that ${\displaystyle 7\sqrt{5}=\frac{a}{b}}$ Where b ≠ 0

Now, ${\displaystyle 7\sqrt{5}=\frac{a}{b}}$

⇒ ${\displaystyle \frac{a}{7b}=\sqrt{5}.}$

Now, ${\displaystyle \frac{a}{7}}$ is a rational number since the product of two rational numbers is a rational number.

The above will imply that ${\displaystyle \sqrt{5}}$ is a rational number. But ${\displaystyle \sqrt{5}}$ is an irrational number.

∴ Our assumption is wrong.

Thus, we conclude that ${\displaystyle 7\sqrt{5}}$ is irrational.