Proving the Equation Involving the Square Root of (10-2√5)

Question

Title: Proving the Equation I am trying to prove the equation:

$$\sqrt{10 - 2\sqrt{5}} = (\sqrt{5} + 1) \sqrt{5 - 2\sqrt{5}}$$

I have tried simplifying both sides, but I am unable to show that they are equal. Can someone help me prove this equation?

Thank you!

Answer 1

$(\sqrt{5}+1)^2 = 5 + 2\sqrt{5} + 1 = 6+2\sqrt{5}$, so the RHS can be written as $$\sqrt{(\sqrt{5}+1)^2(5-2\sqrt{5})} = \sqrt{(6+2\sqrt{5})(5 - 2\sqrt{5})}$$ $$= \sqrt{30 -2\sqrt{5} -20}$$ $$=\sqrt{10-2\sqrt{5}}$$

Answer 2

Note that $10-2\sqrt{5}=(6+2\sqrt{5})(5-2\sqrt{5})$ and $(6+2\sqrt{5})$ is equal to $(1+\sqrt{5})^2$. The rest follows.