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Just a quick question i'm currently confused, would be grateful if you anyone could provide full working out.

"Without resorting to the use of a calculator or computer, find a simpler representation for each of the numbers below:"

$\sqrt{2+\sqrt3} - \sqrt{2-\sqrt3}$

Extend to find: $\sqrt{2+\sqrt3} + \sqrt{2-\sqrt3}$

Generalise to find: $\sqrt{a+\sqrt b} \pm \sqrt{a-\sqrt b}$

Thanks..

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  • $\begingroup$ Welcome to MSE. Please use MathJax. $\endgroup$ – José Carlos Santos Feb 5 '18 at 9:59
  • $\begingroup$ I was just going to suggest some corrections but it seems that you have managed most yourself. You can use \pm for the plus / minus symbol. $\endgroup$ – badjohn Feb 5 '18 at 10:02
  • $\begingroup$ See en.wikipedia.org/wiki/Nested_radical $\endgroup$ – lhf Feb 5 '18 at 10:07
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Notice that $2\cdot2=\sqrt3^2+1^2$ and

$$\sqrt{\frac{2(\sqrt3\pm1)^2}2}=\frac{\sqrt3\pm1}{\sqrt2}.$$

Hence your expressions are $\sqrt 6$ and $\sqrt2$.

To generalize, write

$$a\pm\sqrt b=(\alpha\pm\beta\sqrt b)^2=\alpha^2+\beta^2b\pm2\alpha\beta\sqrt b$$ and identify.

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  • $\begingroup$ Thanks for the help, could you please elaborate on the generalization, still quite confused. $\endgroup$ – JustyourAverage Feb 6 '18 at 9:47
  • $\begingroup$ @JustyourAverage: you need to identify the $a$ and $\sqrt b$ terms and solve for $\alpha,\beta$. You will see that a condition is required to obtain a simpler solution. $\endgroup$ – Yves Daoust Feb 6 '18 at 9:51

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