How do you simplify (4 - sqrt 7) /(3+sqrt7)?

Jul 25, 2018

$\frac{19}{2} - \frac{7}{2} \sqrt{7}$

Explanation:

$\text{multiply the numerator/denominator by the conjugate of}$
$\text{the denominator}$

$\text{the conjugate of "3+sqrt7" is } 3 \textcolor{red}{-} \sqrt{7}$

$= \frac{\left(4 - \sqrt{7}\right) \left(3 - \sqrt{7}\right)}{\left(3 + \sqrt{7}\right) \left(3 - \sqrt{7}\right)}$

$= \frac{12 - 7 \sqrt{7} + 7}{9 - 7}$

$= \frac{19 - 7 \sqrt{7}}{2} = \frac{19}{2} - \frac{7}{2} \sqrt{7}$

$\frac{19 - 7 \setminus \sqrt{7}}{2}$

Explanation:

$\setminus \frac{4 - \setminus \sqrt{7}}{3 + \setminus \sqrt{7}}$

$= \setminus \frac{\left(4 - \setminus \sqrt{7}\right) \left(3 - \setminus \sqrt{7}\right)}{\left(3 + \setminus \sqrt{7}\right) \left(3 - \setminus \sqrt{7}\right)}$

$= \setminus \frac{12 - 3 \setminus \sqrt{7} - 4 \setminus \sqrt{7} + 7}{{3}^{2} - {\left(\setminus \sqrt{7}\right)}^{2}}$

$= \setminus \frac{19 - 7 \setminus \sqrt{7}}{9 - 7}$

$= \setminus \frac{19 - 7 \setminus \sqrt{7}}{2}$