Exercises: Making Denominators Rational

\displaystyle \frac{\sqrt{2}-\sqrt{125}}{\sqrt{5}}

=

\displaystyle \frac{\sqrt{2}-\sqrt{125}}{\sqrt{5}}

=

\displaystyle \frac{\left(\sqrt{2}-\sqrt{125}\right)\cdot\sqrt{5}}{\sqrt{5}\cdot\sqrt{5}}

=

\displaystyle \frac{\sqrt{10}-\sqrt{625}}{5}

=

\displaystyle \frac{\sqrt{10}-25}{5}

=

\displaystyle \frac{\sqrt{10}}{5}-5

$\displaystyle \frac{1}{\sqrt{2}}$	$=$	$\displaystyle \frac{1}{\sqrt{2}}$
	↓	Expand the fraction by $\sqrt2$ .
	$=$	$\displaystyle \frac{1\cdot\sqrt{2}}{\sqrt{2}\cdot\sqrt{2}}$
	$=$	$\displaystyle \frac{\sqrt{2}}{2}$

Do you have a question?

$\displaystyle \frac{\sqrt{2}-\sqrt{125}}{\sqrt{5}}$	$=$	$\displaystyle \frac{\sqrt{2}-\sqrt{125}}{\sqrt{5}}$
	↓	Expand the fraction by $\sqrt{5}$ .
	$=$	$\displaystyle \frac{\left(\sqrt{2}-\sqrt{125}\right)\cdot\sqrt{5}}{\sqrt{5}\cdot\sqrt{5}}$
	↓	Multiply.
	$=$	$\displaystyle \frac{\sqrt{10}-\sqrt{625}}{5}$
	↓	Take the square root.
	$=$	$\displaystyle \frac{\sqrt{10}-25}{5}$
	$=$	$\displaystyle \frac{\sqrt{10}}{5}-5$