Express $w$ and $1/w$ for $w=\frac {\sqrt2+\sqrt3}{\sqrt5-\sqrt3}$ in the simplest form with a rational denominator

Question

If $w = \frac {\sqrt2+\sqrt3}{\sqrt5-\sqrt3} $

Express the following in the simplest form (with a rational denominator)

i) $w$

ii) $\frac1w$

I'm confused about (ii) question :/ pls help me.

Answer 1

If $w = \frac {\sqrt 2 + \sqrt 3}{\sqrt 5-\sqrt 3}$, then $\frac 1w=\frac {\sqrt 5-\sqrt 3}{\sqrt 2 + \sqrt 3}$ If you can do i, you should be able to do ii. The process is the same.

Answer 2

$w=\frac {\sqrt2+\sqrt3}{\sqrt5-\sqrt3}\frac {\sqrt5+\sqrt3}{\sqrt5+\sqrt3}=\frac{\sqrt10 + \sqrt15-\sqrt6-9}{16}$

$\frac{1}{w}=\frac {\sqrt5-\sqrt3}{\sqrt2+\sqrt3}\frac {\sqrt2-\sqrt3}{\sqrt2-\sqrt3}=\frac{-\sqrt10 - \sqrt15+\sqrt6+9}{5}$

Answer 3

\begin{align*} w &=\frac{\sqrt{2}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} \\ &=\frac{(\sqrt{2}+\sqrt{3})(\sqrt{5}+\sqrt{3})} {(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})} \\ &=\frac{3+\sqrt{6}+\sqrt{10}+\sqrt{15}}{5-3} \\ &=\frac{1}{2} (3+\sqrt{6}+\sqrt{10}+\sqrt{15}) \\ \frac{1}{w} &=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{2}+\sqrt{3}} \\ &=\frac{(\sqrt{5}-\sqrt{3})(\sqrt{3}-\sqrt{2})} {(\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2})} \\ &=\frac{\sqrt{15}-\sqrt{10}+\sqrt{6}-3}{3-2} \\ &=\sqrt{15}-\sqrt{10}+\sqrt{6}-3 \end{align*}