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

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

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

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

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.