SOLUTION: how do you simplify the 8th root of 16x^4 y^12

Question 249706: how do you simplify the 8th root of 16x^4 y^12
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
the 8th root of

First we need to know what this means. "the 8th root of means
Second, it helps if we understand how to write this using fractional exponents.

Next, we should recognize that is a single term (with many factors). Since it is a single term, the following rule for exponents applies:

(This looks similar to the Distributive Property but it is not the Distributive Property.) We can use this to raise each factor to the 1/8th power:

Since we can rewrite this as:

Now we can use another rule for exponents, , to simplify the expression:

which simplifies to:

(Can you now see why we changed into ?)
Since all the exponents have the same denominator they all represent the same root: square root. We can factor out a 1/2 from each exponent using the earlier property (in the other direction):

To finish this we will write it will a radical:

The only thing left here is to simplify, if possible. Simplifying square roots involves finding perfect square factors, if any. We do have a perfect square factor, , which we can factor out:

(Note that we can't say because square roots are positive but y does not have to be positive. So we use to ensure that we end up with a positive number.)
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