<s>[INST] Proving $\sqrt{2a + 2\sqrt{a^2 - b}}=\sqrt{a-\sqrt{b}} + \sqrt{a+\sqrt{b}}$, where $a\geq 0,\ b\geq 0$. Who can help me to prove this equation?
$$\sqrt{2a + 2\sqrt{a^2 - b}}=\sqrt{a-\sqrt{b}} + \sqrt{a+\sqrt{b}}$$ Where, $$a\geq 0,\ b\geq 0$$
 [/INST] We have
$$\left(\sqrt{a-\sqrt{b}}+\sqrt{a+\sqrt{b}}\right)^2 = a-\sqrt{b}+a+\sqrt{b} +2\sqrt{a^2-b} = 2a+2\sqrt{a^2-b}$$
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