Question

How do you simplify `(2+5sqrt3)/(-4+4sqrt2)`?

Answer 1

`(2+2sqrt2+5sqrt3+5sqrt6)/4`

Explanation:

In surd form we must not have surds (roots) in the denominator.
We use the fact that `(a+b)(a-b)=a^2-b^2`

The denominator factorises to `4(-1+sqrt2)`
So we multiply both numerator and denominator by the same thing `(-1-sqrt2)`

`(2+5sqrt3)/(4(-1+sqrt2))` =`(2+5sqrt3)/(4(-1+sqrt2))*((-1-sqrt2)/(-1-sqrt2))`
Numerator`(2+5sqrt3)(-1-sqrt2)=-2-2sqrt2-5sqrt3-5sqrt6`
or `-(2+2sqrt2+5sqrt3+5sqrt6`)

Denominator `4(-1+sqrt2)(-1-sqrt2)=4(1-2)=-4`

Hence answer!