Question

Prove that $5\sqrt{2}+3$ is an irrational number.

Answer 1

Assume, $5\sqrt{2}+3$ is a rational number.

Then it should be represented in the form of $\frac{p}{q}$.

$\Rightarrow 5\sqrt{2}+3=\frac{p}{q}$

$5\sqrt{2}=\frac{p}{q}-3$

$\sqrt{2}=\frac{a-3b}{5b}$

$\sqrt{2}$ is irrational.

$\frac{a-3b}{5b}$ is rational.

Irrational number can't be equated to rational number.

$\therefore 5\sqrt{2}+3$ is an irrational number.