SOLUTION: 32x^6y^9/z^8 all under a third root sign

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Question 276413: 32x^6y^9/z^8
all under a third root sign
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
root%283%2C+32x%5E6y%5E9%2Fz%5E8%29
Simplifying a cube root involves finding perfect cubes. Since there is a denominator we will also want to rationalize the denominator. I'm going to make the denominator a perfect cube at the start. This will take care of the rationalizing denomiinator before we start. Since any exponent that is divisible by 3 is a perfect cube, all I need to do is change z%5E8 to z%5E9:
root%283%2C+%2832x%5E6y%5E9%2Fz%5E8%29%28z%2Fz%29%29
which simplifies to:
root%283%2C+32x%5E6y%5E9z%2Fz%5E9%29
Now we can start finding perfect cubes:
root%283%2C+2%5E3%2A4%28x%5E2%29%5E3%28y%5E3%29%5E3z%2F%28z%5E3%29%5E3%29
Using the properties of radicals we can separate out all the perfect cubes:

Replacing the cube roots of the perfect cubes we get:
%282%2Aroot%283%2C+4%29%2Ax%5E2%2Ay%5E3%2Aroot%283%2C+z%29%29%2Fz%5E3
Rearranging the factors in the numerator and combining the remaining radicals we get:
%282x%5E2y%5E3%2Aroot%283%2C+4z%29%29%2Fz%5E3