Question

How do you rationalize the denominator and simplify `1/ (sqrt 3 - sqrt 5)`?

Answer 1

`-1/2(sqrt3+sqrt5)`

Explanation:

What we have to do here is to multiply the numerator and denominator by the`color(blue)" conjugate"" of the denominator"`

The conjugate here is `sqrt3+sqrt5`

Note that the radicals remain unchanged ,while the 'sign' changes.

If `sqrt3-sqrt5" then conjugate" =sqrt3+sqrt5`

`rArr1/(sqrt3-sqrt5)=(1(sqrt3+sqrt5))/((sqrt3-sqrt5)(sqrt3+sqrt5))`

Consider the denominator.

`(sqrt3-sqrt5)(sqrt3+sqrt5)" and expand using FOIL"`

`=(sqrt3)^2+cancel(sqrt5 .sqrt3)-cancel(sqrt5 .sqrt3)-(sqrt5)^2`

`=3-5=-2" which is rational"`

`rArr1/(sqrt3-sqrt5)=(sqrt3+sqrt5)/(-2)=-1/2(sqrt3+sqrt5)`