Question

If $p$ and $q$ are the positive prime number, then prove that $\sqrt{p}+\sqrt{q}$ is an irrational number.

Answer 1

Let us asume $\sqrt{p}+\sqrt{q}$ are rational.

Given that $p$ and $q$ are prime positive integers.

$\sqrt{p}+\sqrt{q}=\frac{a}{b}$

Squaring on both sides, we get

$p+q+2\sqrt{pq}={\left(\frac{a}{b}\right)}^2$

$\sqrt{pq}=\frac{1}{2}[{\left(\frac{a}{b}\right)}^2-p-q]-\left(i\right)$

We know that $p$ & $q$ are prime positive numbers

$\Rightarrow \sqrt{p}$, $\sqrt{q}$ and $\sqrt{pq}$ are irrational

So our assumption $\sqrt{p}+\sqrt{q}$ are rational is incorrect.

$\Rightarrow \sqrt{p}+\sqrt{q}$ is a irrational number if $p,q$ are prime positive numbers