Question

Prove that $3+\sqrt{7}$ is irrational number.

Answer 1

We have to prove that $3+\sqrt{7}$ is irrational.

Let us assume the opposite, that $3+\sqrt{7}$ is rational.

Hence $3+\sqrt{7}$ can be written in the form $\frac{a}{b}$ where $a$ and $b$ are co-prime and $b\ne 0$

Hence $3+\sqrt{7}=\frac{a}{b}$

$\Rightarrow \sqrt{7}=\frac{a}{b}-3$

$\Rightarrow \sqrt{7}=\frac{a-3b}{b}$

where $\sqrt{7}$ is irrational and $\frac{a-3b}{b}$ is rational.

Since,rational$\ne$ irrational.

This is a contradiction.

$\therefore$ Our assumption is incorrect.

Hence $3+\sqrt{7}$ is irrational.

Hence proved.