Assume that $\sqrt{p},+\sqrt{q}$ is rational. So,

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

$\mathrm{p}+\mathrm{q}+2\sqrt{\mathrm{pq}},=\frac{{\mathrm{a}}^2}{{\mathrm{b}}^2}$

$\sqrt{\mathrm{pq}},=\frac{1}{2}(\frac{{\mathrm{a}}^2}{{\mathrm{b}}^2}\text{–}\mathrm{p}-\mathrm{q})$

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

Therefore, it is true that $\sqrt{\mathrm{p}},+\sqrt{\mathrm{q}}$ is irrational.