Question

How do you simplify `\frac { \sqrt { 2- \sqrt { 3} } } { \sqrt { 2+ \sqrt { 3} } }`?

Answer 1

`-(13-4sqrt(3))/13`

Explanation:

Square top and bottom, to remove both square roots:
`(sqrt(2-sqrt(3))/sqrt(2+sqrt(3)))^2=(sqrt(2-sqrt(3)))^2/(sqrt(2+sqrt(3)))^2=(2-sqrt(3))/(2+sqrt(3))`

Now, to rationalise the denominator by multiplying the top and bottom by the negative of the square root:
`((2-sqrt(3))(2-sqrt(3)))/((2+sqrt(3))(2-sqrt(3)))=(4-2sqrt(3)-2sqrt(3)+9)/(4+2sqrt(3)-2sqrt(3)-9)=(13-4sqrt(3))/(-13)=-(13-4sqrt(3))/13`

Answer 2

`= (2 - sqrt3)`

Explanation:

`sqrt(2 - sqrt3)/sqrt(2+sqrt3)` => Rationalize the denominator:

`= sqrt(2 - sqrt3)/sqrt(2+sqrt3) * sqrt(2 - sqrt3)/sqrt(2-sqrt3)`

`= (2 - sqrt3)/sqrt(4-sqrt9)`

`= (2 - sqrt3)/sqrt(4 - 3)`

`= (2 - sqrt3)/sqrt1`

`= (2 - sqrt3)`