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..
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.