SOLUTION: This is one of my problems i am having trouble understanding how to solve. Can someone help me understand.? Rationlize the denominator. all variables represent the positive real

Algebra ->  Radicals -> SOLUTION: This is one of my problems i am having trouble understanding how to solve. Can someone help me understand.? Rationlize the denominator. all variables represent the positive real      Log On


   

Question 594403: This is one of my problems i am having trouble understanding how to solve. Can someone help me understand.?
Rationlize the denominator. all variables represent the positive real numbers.
(12t^3)^(1/3)/(54t^2)^(1/3)
here is the link on how my problem looks like: http://www.wolframalpha.com/input/?i=%2812x%5E3%29%5E%281%2F3%29%2F%2854x%5E2%29%5E%281%2F3%29
Thank you.
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
root%283%2C+12t%5E3%29%2Froot%283%2C+54t%5E2%29%29
As usual in Math there is more than one way to do this. My solution starts with using a property of radicals, root%28a%2C+p%29%2Froot%28a%2C+q%29+=+root%28a%2C+p%2Fq%29, to merge the two cube roots into one big one:
root%283%2C+12t%5E3%2F54t%5E2%29%29

By doing this we now have a fraction inside the cube root which will reduce. (This is why I used the property to merge the cube roots.)
root%283%2C+%286%2A2%2At%5E2%2At%29%2F%286%2A9%2At%5E2%29%29

leaving:
root%283%2C+%282t%29%2F9%29

Now we must address rationalizing the denominator. To do this we will first make the denominator a perfect cube (8, 27, 64, 125, etc.) The easiest one to create out of the 9 we have is 27:
root%283%2C+%28%282t%29%2F9%29%283%2F3%29%29
root%283%2C+%286t%29%2F27%29
Now we can use the earlier property, in the opposite direction to split the cube root in two:
root%283%2C+6t%29%2Froot%283%2C+27%29
And finally we replace the cube root of the perfect cube in the denominator:
root%283%2C+6t%29%2F3