Prove $\sqrt{3} + \sqrt{5}$ is irrational If it is assumed that $\sqrt{3}$ is known to be irrational (not the case for $\sqrt{5}$), then prove that $\sqrt{3}+\sqrt{5}$ is irrational.
My approach:
Assume that $\sqrt{3}+\sqrt{5}$ is rational. Then there exist coprime integers $p$ and $q$ so that $\frac{p}{q}$ is rational and $\frac{p}{q}=\sqrt{3}+\sqrt{5}$. Thus $(\sqrt{3}+\sqrt{5})^2=2(4+\sqrt{15})=\frac{p^2}{q^2}$, which implies that $p^2$ is even, so $p$ is also even. Let $p:=2m$ for some integer $m$, then $p^2=4m^2$. Thus $q^2(4+\sqrt{15})=2m^2$, which implies that $q^2$ is even, and so $q$ is even. But this contradicts that $p$ and $q$ are coprime, and we arrive at a contradiction.

My way of proving this does not use the fact that $\sqrt{3}$ is
  irrational. Please let me know if my proof is correct, and how to use
  the above mentioned fact.


Let $x=\sqrt 3 + \sqrt 5$. Then:
$$(x - \sqrt 3)^2 = 5$$
$$x^2 - 2 \sqrt 3 x + 3 = 5$$
$$x^2 - 2 = 2 \sqrt 3 x$$
$$(x^2 - 2)^2 = 12 x^2$$
$$x^4 -16 x^2 + 4 = 0$$
By the rational root theorem, if the last equation had rational roots then they would be integer divisors of $4$. But none of $\pm1, \pm2,\pm4$ are roots, so $x$ must be irrational.
