SOLUTION: Please help me solve this problem: (squareroot)x^2/18 both the numerator and denominator are under one square root sign. the instructions say to rationalize the denominator.

Click here to see ALL problems on Square-cubic-other-roots

Question 251882: Please help me solve this problem: (squareroot)x^2/18 both the numerator and denominator are under one square root sign.
the instructions say to rationalize the denominator.
Becasue 18's square root is 4.24 I am unsure this is the right answer (x/4.24) I believe I may be missing something. Should I factor out the denominator?
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

In the first place 18's square root is NOT 4.24. 4.24 is only an approximation of the square root of 18. You cannot exactly represent the square root of 18 with a decimal fraction. If you could exactly represent the square root of 18 with some decimal fraction, then the square root of 18 would have to be rational -- that is because any decimal fraction with a finite number of decimal places can be represented exactly by the quotient of two integers. But we can prove that 18 is not rational...

The first thing to do is to find the prime factorization of 18, namely 2 times 3 times 3. Take two factors of 3 out of the radical and leave one of them on the outside, hence:

Now your problem looks like:

So far, so good, but we still have a radical (irrational number) in the denominator. It would be tidy if we were able to multiply the denominator by , but that would change the value of the fraction. However, we can multiply the entire fraction by 1 and not change its value. So let's multiply by 1 in the form of , thus:

Now you have an equivalent representation of the original expression with a rational number denominator.

See my lesson on Rationalizing Denominators:

http://www.algebra.com/algebra/homework/Radicals/rationalizingdenominators1.lesson

John