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I can't figure out how to solve this, I don't really know how or where to find the solution, even the same problems just to figure out, but I didn't find anything. Please help. $$ \sqrt {5 - \sqrt {13 + \sqrt {48}}} $$

I was thinking just to use a calculator, but I am not sure about this. Any advices, any links would help very much, Thanks!!!

$$ \left(\sqrt {5 - \sqrt {13 + \sqrt{48}}}\right) = {?} $$ How can I solve this?

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    $\begingroup$ What are you trying to figure out, exactly? Whether this number can be written in a simpler form? $\endgroup$
    – Sambo
    Sep 7, 2018 at 12:59
  • $\begingroup$ When you write $\sqrt{4}8$, do you mean $\sqrt{4}\times 8 =16$ or $\sqrt{48}$? $\endgroup$
    – Kusma
    Sep 7, 2018 at 13:00
  • $\begingroup$ @Kusma \sqrt {48}} Sorry i fixed $\endgroup$
    – Asylzat
    Sep 7, 2018 at 13:01
  • $\begingroup$ Considering the user mentions using a calculator, they apparently want to evaluate the expression to get a decimal. $\endgroup$
    – rschwieb
    Sep 7, 2018 at 13:02
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    $\begingroup$ First write $\sqrt{48} = 2\sqrt{12}$, and later write $\sqrt{12} = 2 \sqrt{3}$. $\endgroup$ Sep 7, 2018 at 13:06

3 Answers 3

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Hint to get you started: $$\sqrt{13 + \sqrt{48}} = \sqrt{(\sqrt{12})^2 + (\sqrt{1})^2 + 2\sqrt{12}\sqrt{1}}$$

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  • $\begingroup$ $\sqrt{5 - \sqrt{12} - 1}$ = $\sqrt{3 - \sqrt{12} + 1}$ = $\sqrt3 - 1$ $\endgroup$
    – Asylzat
    Sep 7, 2018 at 13:28
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    $\begingroup$ Thank you very much, thats really easy, you are the best $\endgroup$
    – Asylzat
    Sep 7, 2018 at 13:29
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The following method is often useful in similar exercises:

Find $a,b$ such that $\left(a+b\sqrt3\right)^2=13+4\sqrt{3}.$ This leads to the system $$\left \{ \begin{array}{1} a^2+3b^2&=13\\ ab&=2 \end{array}\right.$$ with two solutions $(a,b)=(\pm1,\pm2).$ Consequently $\sqrt{13+\sqrt{48}}=1+2\sqrt3.$

Once again this method: $$(c+d\sqrt3)^2=5-\left(1+2\sqrt3\right)$$ leads to $$\left \{ \begin{array}{1} c^2+3d^2&=4\\ cd&=-1 \end{array}\right.$$

and we have $(c,d)=(\pm1,\mp1)$.
We can write $$\sqrt{5-\sqrt{13+\sqrt{48}}}=-1+\sqrt3,$$ as the square root is non-negative.

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Since we have $48=3\times 16$, we can write $$\sqrt{5-\sqrt{13+4\sqrt{3}}}$$

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