# How do you rationalize the denominator and simplify sqrt15/(sqrt15-sqrt13)?

Mar 10, 2018

See a solution process below:

#### Explanation:

To rationalize the denominator multiply the fraction by the appropriate form of $1$

$\frac{\textcolor{red}{\sqrt{15}} + \textcolor{red}{\sqrt{13}}}{\textcolor{red}{\sqrt{15}} + \textcolor{red}{\sqrt{13}}} \times \frac{\sqrt{15}}{\sqrt{15} - \sqrt{13}} \implies$

$\frac{\textcolor{red}{\sqrt{15}} \sqrt{15} + \textcolor{red}{\sqrt{13}} \sqrt{15}}{\textcolor{red}{\sqrt{15}} \sqrt{15} - \textcolor{red}{\sqrt{15}} \sqrt{13} + \textcolor{red}{\sqrt{13}} \sqrt{15} - \textcolor{red}{\sqrt{13}} \sqrt{13}} \implies$

$\frac{15 + \sqrt{\textcolor{red}{13} \cdot 15}}{15 - 0 - 13} \implies$

$\frac{15 + \sqrt{195}}{2}$

Mar 10, 2018

$\frac{15 + \sqrt{195}}{2}$

#### Explanation:

$\frac{\sqrt{15}}{\sqrt{15} - \sqrt{13}}$

:.color(magenta)((sqrt15+sqrt13)/(sqrt15+sqrt13)=1

:.sqrt15/(sqrt15-sqrt13)xxcolor(magenta)((sqrt15+sqrt13)/(sqrt15+sqrt13)

:.color(magenta)(=sqrt15xxsqrt15=15

$\therefore = \frac{\sqrt{15} \left(\sqrt{15} + \sqrt{13}\right)}{\left(\sqrt{15} - \sqrt{13}\right) \left(\sqrt{15} + \sqrt{13}\right)}$

$\therefore = \frac{\sqrt{15} \left(\sqrt{15} + \sqrt{13}\right)}{15 - 13}$

$\therefore = \frac{\sqrt{15} \left(\sqrt{15} + \sqrt{13}\right)}{2}$

$\therefore = \frac{15 + \sqrt{13} \sqrt{15}}{2}$

$\therefore = \frac{15 + \sqrt{195}}{2}$