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

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

Answer

Let us assume $5 + 3\sqrt{2}$ is a rational number.

$(5 + 3\sqrt{2}) = \frac{p}{q}$  where $q \ne 0$ and p and q are integers.

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

$\sqrt{2}$ is a rational number as RHS is rational

This contradicts the given fact that $\sqrt{2}$ is irrational.

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