SOLUTION: Use rational exponents to simplify. Write the answer in radical notation, if possible. exponent of 12 SQRT (a^10)

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Question 352392: Use rational exponents to simplify. Write the answer in radical notation, if possible.

exponent of 12 SQRT (a^10)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
For this problem you need to understand:
  • The relationship between radicals and rational/fractional exponents: q%5E%28a%2Fb%29+=+root%28b%2C+q%5Ea%29+=+%28root%28b%2C+q%29%29%5Ea
  • Square roots are "second roots". IOW: sqrt%28x%29 is the same as root%282%2C+x%29.

So 12sqrt%28a%5E10%29 written with rational exponents would be:
12%2A%28a%5E10%29%5E%281%2F2%29
Using the rule for exponents, %28a%5Ep%29%5Eq+=+a%5E%28%28p%2Aq%29%29, we can simplify the "a" part of the above:
12%2Aa%5E%28%2810%2A%281%2F2%29%29%29
which simplifies to
12a%5E5

There is only one thing left. All square roots are supposed to be positive or zero, including sqrt%28a%5E10%29. And 12 times a positive number or zero, like 12sqrt(a^10), will be positive or zero. However, our answer at this point is not necessarily positive or zero. In fact, if "a" is negative then 12a%5E5 will be negative, too. Since "a" could be negative (there's no reason it can't be), we cannot use 12a%5E5 for the answer. We must somehow ensure that our answer must be positive or zero. There are a variety of expressions we could use. Probably the preferred one is found as follows:
12a%5E5
12a%5E4%2Aa
12a%5E4%2Aabs%28a%29
This expression will always be positive or zero no matter what "a" is.