Simplification of Irrational Numbers

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..

• Welcome to MSE. Please use MathJax. – José Carlos Santos Feb 5 '18 at 9:59
• 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. – badjohn Feb 5 '18 at 10:02
• – lhf Feb 5 '18 at 10:07

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 $\sqrt2$.
$$a\pm\sqrt b=(\alpha\pm\beta\sqrt b)^2=\alpha^2+\beta^2b\pm2\alpha\beta\sqrt b$$ and identify.
• @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. – Yves Daoust Feb 6 '18 at 9:51