Write $\sqrt{m} + \sqrt{n} + \sqrt{p} = q$ then, subtracting $\sqrt{p}$ from both sides and squaring we obtain \[m + n + 2\sqrt{mn} = q^2 + p - 2q\sqrt{p}.\] A bit of algebra tells us that $\sqrt{mn} + q \sqrt{p}$ is rational, lets call it $r$. Subtracting $q\sqrt{p}$ from both sides and squaring again yields \[mn = r^2 - 2qr\sqrt{p} + q^2p\] and hence we conclude that $\sqrt{p}$ is rational. By symmetry they are all rational.