Simplify the following product into one radical. Then, extract the perfect nth roots inside the radical.

[supercalifragilicious007]

Simplify the following product into one radical. Then, extract the perfect nth roots inside the radical.

 

[Dr.Peterson]

View attachment 34735

The wording of your title is odd; is that what the problem said, exactly? If not, please quote the instructions, using an image if necessary.

My impression is that you should rewrite the expression in exponential form and then simplify; but what the second sentence means is very unclear, as there are no nth roots inside a radical, even after simplifying.

 

[Otis]

extract the [perfect roots]

Hi SCFL007. After writing the given product as a single radical (a 6th root), the resulting radicand has no perfect root. Neither does the radical itself. We could extract only some 6th roots of x^13 and of y^19, and the numerical coefficient cannot be simplified to an Integer. ?

Please respond to Dr. Peterson's question about the instructions. Thanks!
[imath]\;[/imath]

 

[topsquark]

Simplify the following product into one radical. Then, extract the perfect nth roots inside the radical.View attachment 34735

It sounds like what they are asking is to find the "least common radical" or some-such. We have a square root (n = 2) and a cube root (m = 3) so the LCM is 6. So
[imath]\left ( \sqrt{3 x y^5} \right ) \left ( \sqrt[3]{4 x^5 y^2} \right )[/imath]

[imath]= \left ( \sqrt[6]{ (3 x y^5 )^3 } \right ) \left ( \sqrt[6]{ (4 x^5 y^2 )^2 } \right )[/imath]

Now combine these and do any simplification needed.

-Dan

 

[Steven G]

\(\displaystyle \sqrt{3xy^5}=3^{1/2}x^{1/2}y^{5/2}\)
Now do the same with the 2nd factor.
Can you combine anything?