Savanah Boone

2022-07-02

Express $w$ and $1/w$ for $w=\frac{\sqrt{2}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ in the simplest form with a rational denominator
Express the following in the simplest form (with a rational denominator)
i) $w$
ii) $\frac{1}{w}$
I'm confused about (ii) question :/ pls help me.

Jayvion Mclaughlin

Expert

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

klipbodok6

Expert

$\begin{array}{rl}w& =\frac{\sqrt{2}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}\\ & =\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}\\ & =\frac{3+\sqrt{6}+\sqrt{10}+\sqrt{15}}{5-3}\\ & =\frac{1}{2}\left(3+\sqrt{6}+\sqrt{10}+\sqrt{15}\right)\\ \frac{1}{w}& =\frac{\sqrt{5}-\sqrt{3}}{\sqrt{2}+\sqrt{3}}\\ & =\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{3}-\sqrt{2}\right)}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{3}-\sqrt{2}\right)}\\ & =\frac{\sqrt{15}-\sqrt{10}+\sqrt{6}-3}{3-2}\\ & =\sqrt{15}-\sqrt{10}+\sqrt{6}-3\end{array}$