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.

We know that the theorem "The product of any irrational number with a rational number is irrational".

So, since we know $3$ is a rational number, $3\sqrt{2}$ is irrational.

$(3=\frac{3}{1};1\ne 0)$

Now we know that $3\sqrt{2}$ is irrational.

$Theorem:$ The sum of a rational number with an irrational number is irrational.

So, since we know $5$ is a rational number, $5+3\sqrt{2}$ is irrational.

$(5=\frac{5}{1};1\ne 0)$

Thus we proved that $5+3\sqrt{2}$ is an irrational number.