SOLUTION: Simplify the following radical: cubic root 3 sqrt − 64c^15/cubic root 3 sqrt x^15

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Question 201402: Simplify the following radical: cubic root 3 sqrt − 64c^15/cubic root 3 sqrt x^15
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
If the problem is to simplify %28root%283%2C+-64c%5E15%29%29%2F%28root%283%2C+x%5E15%29%29 then there are some key ideas we need to do this:
  1. An understanding of what roots are (discussed below)
  2. The properties of exponents
    • %28x%5Ey%29%5Ez=x%5E%28y%2Az%29
    • x%5Ey%2Ax%5Ez=x%5E%28y%2Bz%29
    • %28x%5Ey%29%2F%28x%5Ez%29=x%5E%28y-z%29
  3. The properties of roots:
    • root%28n%2C+x%2Ay%29+=+root%28n%2C+x%29+%2A+root%28n%2C+y%29
    • root%28n%2C+%28x%2Fy%29%29+=+%28root%28n%2C+x%29%29%2F%28root%28n%2C+y%29%29

Using the first property of roots above we can separate the numerator into two roots:
%28%28root%283%2C+-64%29%2Aroot%283%2C+c%5E15%29%29%29%2F%28root%283%2C+x%5E15%29%29
Now we can simplify each of these three roots separately. To do so we need to understand what cube (or cubic) roots represent. Cube roots represent the number or expression which can be cubed and result in the expression within the radical.
So root%283%2C1%29 stands for the number or expression which you can cube and get 1. Since %281%5E3%29+=+1 root%283%2C+1%29+=+1. Since 2%5E3+=+8 root%283%2C+8%29+=+2. Since %28x%29%5E3+=+x%5E3 root%283%2C+x%5E3%29+=+x. Similarly root%283%2C+%28x%2By%29%5E3%29+=+x%2By.
So to simplify our three roots we have to answer the following questions:
  • What cubed results in -64?
  • What cubed results in c%5E15?
  • What cubed results in x%5E15?

The first one just requires a little trial and error (or familiarity with the first few perfect cubes) to find that %28-4%29%5E3+=+-64. Therefore root%283%2C+-64%29=+-4.
The other two roots can be found if we understand the first property of exponents above: %28x%5Ey%29%5Ez+=+x%5E%28y%2Az%29. When we want to cube something then z would be 3: %28x%5Ey%29%5E3=x%5E%28y%2A3%29=x%5E%283y%29. To find out what to cube to get x%5E15 we need to solve the equation 3y=15. So y = 5. Therefore %28x%5E5%29%5E3=x%5E%285%2A3%29=x%5E15. So root%283%2C+x%5E15%29=x%5E5. Similarly root%283%2C+c%5E15%29=c%5E5.
We now have the three cube roots we need. Substituting these values into the fraction:
%28%28root%283%2C+-64%29%2Aroot%283%2C+c%5E15%29%29%29%2F%28root%283%2C+x%5E15%29%29
we get
%28%28-4%29%2A%28c%5E5%29%29%2F%28x%5E5%29
or
%28-4c%5E5%29%2F%28x%5E5%29