Barry M. answered • 01/14/20
Professor, CalTech Grad; Many Years Tutoring Math, SAT/ACT Prep, Chem
Generally, in questions like this, the first step is to clear the denominator of sq rt signs. This is done also for complex numbers (that are in the form of a + bi, where i is the imaginary sq rt of -1), by using the known factoring for the difference of 2 squares. You multiply by the conjugate, which in this case will be the same expression as the denominator, but with a changed sign from + to -. Remember (a + b)(a - b) = a2 - b2. Also, since the denominator is multiplied by a certain amount, the numerator must also be multiplied by the same amount in order to maintain the same value of the overall expression.
So, in this problem we will have 6(3 - sqrt 7)/[(3 + sqrt 7)(3 - sqrt 7)]
= 6(3 - sqrt 7)/(9 - 7)
= 9 - 3sqrt7
