SOLUTION: cube root (8m^7n^9/n^2m^2) i got: cube root 8m^5n^7 8 ^ 2 *4 ^ 2*2 final answer: 2cube root m^5n^7 ???? just want to make sure I did it right.

SOLUTION: cube root (8m^7n^9/n^2m^2) i got: cube root 8m^5n^7 8 ^ 2 4 ^ 22 final answer: 2cube root m^5n^7 ???? just want to make sure I did it right.

Question 405428: cube root (8m^7n^9/n^2m^2)
i got:
cube root 8m^5n^7
8
^
2 *4
^
2*2
final answer: 2cube root m^5n^7 ????
just want to make sure I did it right.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Your first step is right on. The radicand (the expression inside the radical) simplifies to:

Next we look for factors of the radicand that are perfect cubes. As you already found, 8 is a perfect cube. But there are more perfect cube factors. Because of the way exponents work, the exponent on a perfect cube is not a perfect cube but a multiple of 3! So , , are all perfect cubes, even though 3, 12 and 300 are not themselves perfect cubes.

So your radicand factored into as many perfect cubes as we can find is:

For reasons that will become clear shortly I like to use the Commutative Property to rearrange the order of the factors so that all the perfect cubes are in front:

Next we use a property of radicals, , to split this cube root of a product into a product of cube roots. We want each perfect cube factor in its own cube root. The factors that are not perfect cubes can all go into one cube root:

The cube roots of the perfect cubes will simplify:

This is the simplified cube root. (Note how the radical is at the end. This is the usual way to write terms like this and it is the reason I put all the perfect cubes n the front earlier.)
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