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I got this question

Find the square root of $12+2\sqrt{6}$ expressing your answer in the form $\sqrt{m}+\sqrt{n}$.

I have no idea what this means and how to go about it.

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  • $\begingroup$ What is a surds? $\endgroup$ – Euler....IS_ALIVE Feb 3 '13 at 23:02
  • $\begingroup$ @Euler....IS_ALIVE Are you serious with that question? $\endgroup$ – Mob Feb 3 '13 at 23:04
  • $\begingroup$ Irrational root in American English. $\endgroup$ – Mob Feb 3 '13 at 23:05
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    $\begingroup$ It was a serious question. In 7 years of undergrad and graduate work, I've never seen that word. I googled it, but I am fairly confident that is an unusual word. Someone correct me if I'm wrong. $\endgroup$ – Euler....IS_ALIVE Feb 3 '13 at 23:06
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    $\begingroup$ @Euler....IS_ALIVE Quite usual, actually. $\endgroup$ – Julien Feb 3 '13 at 23:07
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Let $x = \sqrt{12 + 2\sqrt{6}} = \sqrt{n} + \sqrt{m}$. Then $x^2 = 12 + 2\sqrt 6 = n + m + 2 \sqrt{nm}$.

Find $n$ and $m$ such that $n + m = 12$ and $nm = 6$.

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  • $\begingroup$ Can you expatiate on this? $\endgroup$ – Mob Feb 3 '13 at 23:13
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    $\begingroup$ Surdly you can solve the system of equations Damien has given. $\endgroup$ – Euler....IS_ALIVE Feb 3 '13 at 23:18
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The answers for $n$ and $m$ are

$n = 6+\sqrt{30}$

$m = 6-\sqrt{30}$.

Notice that they are interchangable.

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