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Simplify this:

$10 + 2\sqrt5 - 4\sqrt2\sqrt5 - 4\sqrt2$

Help please! $:'($

It came out like this:

$10 + 2\sqrt5 -4\sqrt10 - 4\sqrt2$

How should I do the next step? Um stuck.

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Try grouping similar surds, using amWhy suggestion. –  CAGT Feb 1 at 19:12
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2 Answers 2

up vote 3 down vote accepted

Note that $4 \sqrt{10} = 4 \sqrt 2\cdot\sqrt 5$, and $10 = 2\sqrt 5\cdot \sqrt 5$.

Hint: FACTOR!

$$\begin{align} 10 + 2\sqrt5 - \color{red}{4\sqrt2}\sqrt5 - \color{red}{4\sqrt2} & = \color{blue}{2\sqrt 5}\cdot \sqrt 5 + \color{blue}{2\sqrt 5} - \color{red}{4\sqrt 2}(\sqrt 5 + 1)\\ \\ & = \color{blue}{2\sqrt 5}\color{purple}{\bf (\sqrt 5 + 1)} - 4\sqrt 2\color{purple}{\bf (\sqrt 5 + 1)}\\ \\ & = \;\;\cdots \end{align}$$

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There is no general notion of "simplest" for such radical expressions. What you wrote could well be considered simplest. Yet another possibility is $\,(\sqrt{10}-4)(\sqrt{10}+\sqrt{2}),\,$ which, by various subjective criteria, is arguably simpler than the product $\,(\sqrt{5}+1)(2\sqrt{5}-4\sqrt{2})\,$ in the other answer.

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