How do you simplify \frac { \sqrt { 7} } { 3+ \sqrt { 11} }?

Mar 29, 2018

(3sqrt7-sqrt77)/(-2 or

-(3sqrt7-sqrt77)/(2

Explanation:

Basically to solve this you need to rationalize the denominator aka multiply with it.

Which is this:

$\frac{\sqrt{7}}{\textcolor{red}{3 + \sqrt{11}}}$

To do this, we use the formula ${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

sqrt7/ (3+sqrt11) * (3-sqrt11)/(3-sqrt11

As you can see, $3 + \sqrt{11}$ turned into $3 - \sqrt{11}$ because we used the formula and now:

 = (sqrt7*(3-sqrt11))/color(blue)(3^2-(sqrt11)^2

This blue indicates where the formula was used.

We did $\left(3 + \sqrt{11}\right) \cdot \left(3 - \sqrt{11}\right)$ which is ${3}^{2} - {\left(\sqrt{11}\right)}^{2}$

And finally solve everything that is left,

(3sqrt7-sqrt77)/(-2 or

-(3sqrt7-sqrt77)/(2