# How to calculate $\sqrt[]{(\sqrt{7-2\sqrt[]{10}} + \sqrt[]{2})\cdot 2\sqrt[]{5}}$ [closed]

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

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

## closed as off-topic by Travis, Watson, user99914, user228113, Daniel W. FarlowMar 30 '16 at 14:28

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• What are you trying to do exactly? This can be algebraically manipulated but it's debatable whether or not any other form you'll get is any better than the current one. – tilper Mar 30 '16 at 12:41
• Next time show efforts – Archis Welankar Mar 30 '16 at 12:42

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

hence

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

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}

:)

• The $\LaTeX$ command for a "centered" dot (multiplication) is \cdot. – hardmath Mar 30 '16 at 12:55
• That has been answered above – Theodoros Mpalis Mar 30 '16 at 13:05
• See the time difference i was typing while the answer was given – Archis Welankar Mar 30 '16 at 13:06
• @hardmath we only use \cdot if we have a multi-variable equation. For example, we won’t write $2a\times b$ because $\times$ looks like another letter, namely $x$. Instead, we write $2ab$ or $2a\cdot b$ but the former is most common, unless $a$ and/or $b$ are represented as two seperate quotients. If we had something like $3\times 4$ then this would be ok, but the expression $3\cdot 4$ could come across as a decimal in certain countries which would arouse confusion – Mr Pie Dec 15 '17 at 6:51