Question

Multiply and simplify by factoring. Assume all expressions under radicals represent nonnegative numbers.

^3sqrt y^4 ^3sqsrt 16y^5=
Sorry, not sure how to write this one out

Answer 1

is it like this?
\[\sqrt[3]{y^4}*\sqrt[3]{16y^5}\]

Answer 2

yeah looks like that, but without the multiplication sign

Answer 3

alright, even though the multiplication sign isnt there, thats what is implied by them being next to each other. Also, the correct name for the root is "cube root" since it has that little 3 on the top. Now lets get started :)
To do this problem, first we need to combine the cube roots. We can do this because they are both the same type of root:
\[\sqrt[3]{y^4(16y^5)}\]
Looking at the y^4 and y^5, we can put those together like so:
\[y^4*y^5 = y^{4+5} = y^9\]
So what we really have on the inside is:
\[\sqrt[3]{16y^9}\]
Now we need to find perfect cubes. If we can find some perfect cubes, then we can pull them out of the cube root.
y^9 turns out to be a perfect cube because:
\[y^9 = (y^3)^3\]
While 16 isnt a perfect cube itself, we can split it into 8*2, and 8 is a perfect cube because:
\[8 = 2^3\]
So after making those changes we have:
\[\sqrt[3]{16y^9} = \sqrt[3]{8*2*(y^3)^3} = \sqrt[3]{2*2^3*(y^3)^3} = 2y^3\sqrt[3]{2}\]

Answer 4

and that would be the final answer, sry for the wall of text lol >.<

Answer 5

you're fine. I appreciate your help :)