SOLUTION: Rationalize the denominator and simplify. All variables represent positive real numbers. square root 11 / square root 33 - square root 5

Algebra ->  Radicals -> SOLUTION: Rationalize the denominator and simplify. All variables represent positive real numbers. square root 11 / square root 33 - square root 5      Log On


   

Question 423052: Rationalize the denominator and simplify. All variables represent positive real numbers. square root 11 / square root 33 - square root 5
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%2811%29%2F%28sqrt%2833%29-sqrt%285%29%29
Rationalizing a two-term denominator takes advantage of the pattern:
%28a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2
Notice that the left side is a product of two-term expressions, like your denominator. And the result, the right side, is made up of nothing but perfect squares.

Your denominator has a "-" between the two terms. SO it will play the role fo (a-b) with the "a" being sqrt%2833%29 and the "b" being sqrt%285%29. To get the prefect squares, we need to multiply the (a-b) by (a+b). And if we multiply the denominator by something we must also multiply the numerator by the same thing. So we will multiply the numerator and denominator by (a+b), i.e. %28sqrt%2833%29%2Bsqrt%285%29%29:

We use the distributive Property to multiply the numerators. In the denominators we know how it will multiply because of the pattern we're using:
%28sqrt%28363%29%2Bsqrt%2855%29%29%2F%28%28sqrt%2833%29%29%5E2-%28sqrt%285%29%29%5E2%29
which simplifies as follows:
%28sqrt%28121%2A3%29%2Bsqrt%2855%29%29%2F%2833-5%29
%28sqrt%28121%29%2Asqrt%283%29%2Bsqrt%2855%29%29%2F28
%2811sqrt%283%29%2Bsqrt%2855%29%29%2F28