How do you simplify `(5-3sqrt6)/(3+4sqrt2)`?

In order to rationalize the denominator, we multiply it by its conjugate (the conjugate of `a+bsqrt(c)` is `a-bsqrt(c)`). The reason why will be apparent as we solve the problem.

The conjugate of the denominator `3+4sqrt(2)` is `3-4sqrt(2)`.

Since we are going to multiply its denominator by `3-4sqrt(2)`, we also have to multiply the numerator by `3-4sqrt(2)`.

Then, we have `(5-3sqrt(6))/(3+4sqrt(2))=(5-3sqrt(6))/(3+4sqrt(2))*(3-4sqrt(2))/(3-4sqrt(2))=((5-3sqrt(6))(3-4sqrt(2)))/((3+4sqrt(2))(3-4sqrt(2)))`.

We can simplify the denominator by using the identity `(a+b)(a-b)=a^2-b^2`.

`((5-3sqrt(6))(3-4sqrt(2)))/((3+4sqrt(2))(3-4sqrt(2)))=(15-9sqrt(6)-20sqrt(2)+12sqrt(12))/(9-32)=-(15-9sqrt(6)-20sqrt(2)+24sqrt(3))/23=(9sqrt(6)+20sqrt(2)-24sqrt(3)-15)/23`

We can simplify the denominator by turning it into a single term by multiplying the fraction by the conjugate of the denominator.

The conjugate of `3+4sqrt2` is found by simply reversing the sign of the second term, which is `3-4sqrt2`.

So, we multiply the numerator and denominator by `3-4sqrt2`.

`(5-3sqrt6)/(3+4sqrt2)*(3-4sqrt2)/(3-4sqrt2)=((5-3sqrt6)(3-4sqrt2))/((3+4sqrt2)(3-4sqrt2))`

Expand both of these by FOILing:

`=(15-(3sqrt6)3+5(-4sqrt2)-3sqrt6(-4sqrt2))/(9+(4sqrt2)3+3(-4sqrt2)+(4sqrt2)(-4sqrt2))`

Simplifying these:

`=(15-9sqrt6-20sqrt2+12sqrt12)/(9+12sqrt2-12sqrt2-16sqrt4)`

Note that `sqrt4=2` and `sqrt12=sqrt(4*3)=sqrt4sqrt3=2sqrt3`.

`=(15-9sqrt6-20sqrt2+12(2sqrt3))/(9-16(2))`

`=(15-9sqrt6-20sqrt2+24sqrt3)/(9-32)`

`=(15-9sqrt6-20sqrt2+24sqrt3)/(-23)`