SOLUTION: Please help with this mulitply or divide the radical expressions the answer should be simplified radical the third root of 5w * the 4th root of 5w?

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Question 414090: Please help with this mulitply or divide the radical expressions the answer should be simplified radical the third root of 5w * the 4th root of 5w?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
root%283%2C+5w%29%2Aroot%284%2C+5w%29
We can't use root%28a%2C+p%29%2Aroot%28a%2C+q%29+=+root%28a%2C+p%2Aq%29 because the property requires the same kind of root and your roots are different.

To multiply these roots together we will replace the radicals with fractional exponents. Cube roots are exponents of 1/3 and 4th roots are exponents of 1/4:
%285w%29%5E%281%2F3%29%2A%285w%29%5E%281%2F4%29
Now we have the product of two powers of 5w. The rule for exponents in this situation is to add the exponents. (And the rule does not change just because the exponents happen to be fractions!) So, adding the exponents we get:
%285w%29%5E%28%281%2F3%29%2B%281%2F4%29%29
But to add 1/3 and 1/4 we need a common denominator. (The rules for adding fractions do not change just because they are exponents.)
%285w%29%5E%28%284%2F12%29%2B%283%2F12%29%29
which simplifies to:
%285w%29%5E%287%2F12%29
Now that the multiplication has been done, thanks to the fractional exponents, we can switch back to radical form:
root%2812%2C+%285w%29%5E7%29
which simplifies as follows:
root%2812%2C+5%5E7w%5E7%29
root%2812%2C+78125w%5E7%29
Since 5 is a prime number there can be no power of 12 factors in 5%5E7 or 78125 so this radical cannot be simplified further.