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$\sqrt{57+40\sqrt2} - \sqrt{57-40\sqrt2}$

I'm completely clueless about how to use the formula $a^2 + 2ab + b^2 = (a+b)^2$ to factor the expressions.

With the help of commenters, I successfully factored the expressions.

$\sqrt{57+40\sqrt{2}} = 5+4\sqrt{2}$ and $\sqrt{57-40\sqrt{2}} = 5-4\sqrt{2}$. $5+4\sqrt{2}-4\sqrt{2}+5=10$.

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  • $\begingroup$ $$(a+b\sqrt2)^2 = a^2+2ab\sqrt2+2b^2 = (a^2+2b^2) +2ab\sqrt2.$$ Can you find two numbers $a$ and $b$ such that $a^2+2b^2=57$ and $2ab=40$? $\endgroup$
    – Rócherz
    Commented Sep 11, 2022 at 8:40
  • $\begingroup$ If you are trying to simplify $x=\sqrt{57+40\sqrt2} - \sqrt{57-40\sqrt2}$, you can try to factor $57+40\sqrt{2}$ as @Rócherz explained. Alternatively, you can compute $x^2$ and use identities. $\endgroup$
    – Taladris
    Commented Sep 11, 2022 at 9:32
  • $\begingroup$ Welcome to MathSE. The reason your question has received down votes is that you have not included your attempt. This tutorial explains how to typeset mathematics on this site. $\endgroup$ Commented Sep 11, 2022 at 9:46
  • $\begingroup$ For future reference, you might want to read our guidelines for how to avoid "I'm completely clueless" questions. $\endgroup$
    – Lee Mosher
    Commented Sep 11, 2022 at 13:14

3 Answers 3

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$\sqrt{57+40\sqrt{2}}=\sqrt{25+40\sqrt{2}+32}=\sqrt{5^2+2\cdot 5\cdot 4\sqrt{2}+(4\sqrt{2})^2}=\sqrt{(5+4\sqrt{2})^2}=5+4\sqrt{2}$

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In general, when asked to simplify an expression of the form $$z:=\sqrt{x+a\surd y}\pm\sqrt{x-a\surd y},$$ the trick is to square it, and see what happens. Thus $z^2=\left(\sqrt{x+a\surd y}\pm\sqrt{x-a\surd y}\right)^2$, and so $$z^2=2x\pm2\sqrt{x^2-a^2y}.$$ As usual in this type of problem, $x^2-a^2y$ is a perfect square: in this case, $57^2-40^2\cdot2=49=7^2$. After taking the square root, we end up with $$z=\sqrt{2\cdot57-2\cdot7}=10.$$

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$$\sqrt{57+40\sqrt2} - \sqrt{57-40\sqrt2} $$ $$\Rightarrow \sqrt{25+32+2.(5).(4\sqrt2}) - \sqrt{25+32-2.(5).(4\sqrt2)}$$ $$\Rightarrow 5+4\sqrt2 - (4\sqrt2-5)$$ $$\Rightarrow 10$$

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    $\begingroup$ Your answer is incorrect. $5 - 4\sqrt{2} = \sqrt{25} - \sqrt{32} < 0$. Therefore, it cannot be the square root of $\sqrt{57 - 40\sqrt{2}}$. $\endgroup$ Commented Sep 11, 2022 at 12:53

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