Decide whether the $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ is rational Working needs to be shown
$\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$
My guess is to multiply by  $\sqrt{\sqrt{5}+3}-\sqrt{\sqrt{5}-2}$ then we have a rational number but is it enough to prove the rationality of a number?

If $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ is rational, then so is $\dfrac{5}{\sqrt{\sqrt{5}+3}-\sqrt{\sqrt{5}-2}}$ and consequently $\sqrt{\sqrt{5}+3}-\sqrt{\sqrt{5}-2}\in \mathbb Q$.
The sum of two rational numbers is a rational number, thus $\left(\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}\right)+\left(\sqrt{\sqrt{5}+3}-\sqrt{\sqrt{5}-2}\right)\in \mathbb Q$.
Proceed.
