Question

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

Answer 1

Assume that $\sqrt{5}+\sqrt{3}$ is rational. So,

$\sqrt{5}+\sqrt{3}=\frac{p}{q}$

$5+3+2\sqrt{15}=\frac{p^2}{q^2}$

$\sqrt{15}=\frac{1}{2}\left(\frac{p^2}{q^2}-8\right)$

Since the RHS of the above equation is rational but $\sqrt{15}$ is an irrational number, so the assumption is wrong.

Therefore, $\sqrt{5}+\sqrt{3}$ is irrational.