SOLUTION: Please help me with this problem: Rationalize the denominator. Assume that all expressions under the radical represent positive numbers. 4√7/√5-√3

Algebra ->  Radicals -> SOLUTION: Please help me with this problem: Rationalize the denominator. Assume that all expressions under the radical represent positive numbers. 4√7/√5-√3      Log On


   

Question 383640: Please help me with this problem: Rationalize the denominator. Assume that all expressions under the radical represent positive numbers. 4√7/√5-√3
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
4sqrt%287%29%2F%28sqrt%285%29-sqrt%283%29%29
Your denominator has two terms. Rationalizing a two term denominator uses the pattern: %28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2. The pattern shows us how a binomial, (a+b) or (a-b), can be turned into an expression of perfect squares, a%5E2-b%5E2!

Your denominator has a minus between the two terms. So it will plat the role of (a-b) with "a" being sqrt%285%29 and "b" being sqrt%283%29. To rationalize this denominator we will multiply the numerator and denominator by (a+b):

In the numerator we will use the distributive Property to multiply. In the denominator we already know we will get a%5E2=b%5E2:

which simplifies as follows:
%284sqrt%2835%29%2B4sqrt%2821%29%29%2F%285-3%29
You can already see that the denominator is rational.
%284sqrt%2835%29%2B4sqrt%2821%29%29%2F2
We can reduce this fraction by factoring out a 2 in the numerator:
%282%282sqrt%2835%29%2B2sqrt%2821%29%29%29%2F2
%28cross%282%29%282sqrt%2835%29%2B2sqrt%2821%29%29%29%2Fcross%282%29
2sqrt%2835%29%2B2sqrt%2821%29
Not only did we rationalize the denominator but we eliminated the fraction entirely!