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$$ \sqrt[]{(\sqrt{7-2\sqrt[]{10}} + \sqrt[]{2})\cdot 2\sqrt[]{5}} $$

I know that the answer is $\sqrt[]{10}$, but how do I calculate it mathematically if I don't have access to a calculator?

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2 Answers

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first note $\sqrt{7-2\sqrt{10}}=\sqrt{5+2-2\sqrt5\sqrt2}=\sqrt{(\sqrt5-\sqrt2)^2}=\sqrt5-\sqrt2$

hence

$$\sqrt[]{\big(\sqrt{7-2\sqrt[]{10}} + \sqrt[]{2}\big)\times2\sqrt[]{5}}=\sqrt5\sqrt2$$

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Hint:

$$\begin{align} \sqrt{7-2\sqrt{10}} & =\sqrt{2+5-2\sqrt{10}} \\ &=\sqrt{(\sqrt{5}-\sqrt{2})^2} \\ \therefore \sqrt{\sqrt{5}\times 2\sqrt{5}} &=\sqrt{10} \end{align}$$

:)

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