How to rationalize the denominator of the fraction (3*sqrt 2 - sqrt 6)/(4*sqrt 2 + sqrt 6)

To rationalize the denominator, we'll have to multiply
both the numerator and denominator by the conjugate of the
denominator:

(3sqrt2 - sqrt6)(4sqrt2 - sqrt6)/(4sqrt2 +
sqrt6)(4sqrt2 - sqrt6)

The product of denominator returns a
difference of two squares:

(3sqrt2 - sqrt6)(4sqrt2 -
sqrt6)/(4^2*2 - 6)

We'll remove the brackets from the
numerator:

(12*2 - 3sqrt12 - 4sqrt12 +
6)/(32-6)

We'll combine like terms inside
brackets:

(30 -
7*2sqrt3)/26

(30-14sqrt3)/26 = 2(15 -
7sqrt3)/26

2(15 - 7sqrt3)/26 = (15 -
7sqrt3)/13

The rationalized fraction is (15 -
7sqrt3)/13.