Question 617255
{{{(7sqrt(3)-5sqrt(2))/(sqrt(48)+sqrt(18))}}}
Before going about rationalizing the denominator, I'm going to simplify the denominator. It will make the numbers smaller and easier to work with.
{{{(7sqrt(3)-5sqrt(2))/(sqrt(16*3)+sqrt(9*2))}}}
{{{(7sqrt(3)-5sqrt(2))/(sqrt(16)*sqrt(3)+sqrt(9)*sqrt(2))}}}
{{{(7sqrt(3)-5sqrt(2))/(4sqrt(3)+3sqrt(2))}}}<br>
Now we'll rationalize. Rationalizing two-term denominators with square roots takes advantage of the {{{(a+b)(a-b) = a^2-b^2}}} pattern. The right side, as you can see is a difference of perfect squares. The left side is a product of two-term expressions. The pattern shows us how to take a two-term expression, an (a+b) or (a-b), and turn it into an expression of perfect squares.<br>
Your denominator is a two-term sum. IOW, an (a+b). To turn it into an expression of perfect squares we need to multiply it by its corresponding (a-b):
{{{((7sqrt(3)-5sqrt(2))/(4sqrt(3)+3sqrt(2)))((4sqrt(3)-3sqrt(2))/(4sqrt(3)-3sqrt(2)))}}}
Multiplying the denominators is easy; just use the pattern. To multiply the numerators we will need to use FOIL:
{{{(7sqrt(3)*4sqrt(3)*7sqrt(3)(-3sqrt(2)) + (-5sqrt(2))*4sqrt(3)+(-5sqrt(2))*(-3sqrt(2)))/((4sqrt(3))^2-(3sqrt(2))^2)}}}
which simplifies as follows:
{{{(28*3+(-21sqrt(6))+(-20sqrt(6))+15*2)/(16*3-9*2)}}}
{{{(84+(-41sqrt(6))+30)/(48-18)}}}
{{{(114+(-41sqrt(6)))/30}}}