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

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Answer 1 · 2018-03-29T19:01:36

$(3sqrt7-sqrt77)/(-2$ or

$-(3sqrt7-sqrt77)/(2$

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

Which is this:

$sqrt7/color(red) (3+sqrt11)$

To do this, we use the formula $a^2-b^2=(a-b) (a+b)$

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

As you can see, $3+sqrt11$ turned into $3-sqrt11$ 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 $(3+sqrt11)*(3-sqrt11)$ which is $3^2-(sqrt11)^2$

And finally solve everything that is left,

$(3sqrt7-sqrt77)/(-2$ or

$-(3sqrt7-sqrt77)/(2$

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