Question

Prove the following are irrational.
$3+2\sqrt{5}$

Answer 1

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

That is, we can find two integers $a$ and $b(b\ne 0)$ such that,

$3+2\sqrt{5}=\frac{a}{b}$, where $a$ and $b$ are co-prime integers.

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

$⟹\sqrt{5}=\frac{a-3b}{2b}$

Since, $a$ and $b$ are integers, thus $\frac{a-3b}{2b}$ is rational.

So, $\sqrt{5}$ is also rational.

But, this contradicts the fact that $\sqrt{5}$ is irrational.

Thus, our assumption is wrong that $3+2\sqrt{5}$ is rational.

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