Question

Prove that $\sqrt{3}+\sqrt{5}$ is an irrational number.

Answer 1

Let $\sqrt{3}+\sqrt{5}$ be a rational number.

Let $p=\sqrt{3}+\sqrt{5}$

On squaring both sides, we have

${(\sqrt{3}+\sqrt{5})}^2=r^2$

$3+5+2\sqrt{15}=r^2$

$\Rightarrow 2\sqrt{15}=r^2-8$

$\Rightarrow \sqrt{15}=\frac{(r^2-8)}{2}$

Since $\frac{(r^2-8)}{2}$ is a rational number while $\sqrt{15}$ is an irrational number.$

Since a rational number cannot be equal to an irrational number . Thus the assumption that $\sqrt{3}+\sqrt{5}$ is a rational no. was wrong.

Hence $\sqrt{3}+\sqrt{5}$ is an irrational no.