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Question 190858: Can you please help me solve this? I have to rationalize (8) over (the square root of 32)
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
You can't rationalize  , because no matter what you do, you will still have an irrational factor somewhere in the expression. What you can do is rationalize the denominator, which is what I suspect you meant all along. The language of mathematics is a very precise thing; please endeavor to use it correctly. By the way, if your teacher/instructor actually taught you to say it that way, shame on him or her -- and you have my permission to quote me, including the emphasis. Having vented my spleen, let's get on with it.
First realize that  , so:
The expression becomes: 
So far, so good, but we still have that pesky  in the denominator. The only thing that will change an irrational square root into a rational number is multiplying the root by itself. However, you can't just multiply the denominator by something other than 1 without changing the value of the denominator and therefore the entire fraction. The solution is to multiply your fraction by 1, but in the form of  , thus:
\left( \frac {sqrt{2}} {sqrt{2}}\right) = \frac{2 sqrt{2}}{2} = sqrt{2})
If you are asking yourself why I didn't just divide 2 by root 2 to get root 2 without all of the folderol, it is because these sorts of problems very seldom come out this nicely. It is essential that you learn the process, without any shortcuts, from the very beginning.
John
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