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#