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?
2026-03-25 12:53:31.1774443211
Decide whether the $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ is rational
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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.