Question

Given that $\sqrt{2}$ is irrational, prove that $(5+3\sqrt{2})$ is an irrational number.

Answer 1

Given that $\sqrt{2}$ is irrational number.
Let $\sqrt{2}=m$
Suppose, $5+3\sqrt{2}$ is a rational number.
So, $5+3\sqrt{2}=\frac{a}{b}(a\ne b,b\ne 0)$
$3\sqrt{2}=\frac{a}{b}-5$
$3\sqrt{2}=\frac{a-5b}{b}$
or
$\sqrt{2}=\frac{a-5b}{3b}$
So, $\frac{a-5b}{3b}=m$
But $\frac{a-5b}{3b}$ is rational number, so m is rational number which contradicts the fact that m =
$\sqrt{2}$ is irrational number.
So, our supposition is wrong.
Hence, $5+3\sqrt{2}$ is also irrational.