Algebra help! Simplify completely?

CubeRoots (like square roots or any other level of roots) can be "distributed" over products.

CubeRoot(a) * CubeRoot(b) =

CubeRoot(ab)

Multiply (x^6y^4 / 3z^3)

by (81x^9y^2-10z^6)

*** you are missing the exponent for y in the second cubeRoot **

I am using 2 for now, but you have to do the problem with the real one in your problem

and you get [(27x^15 y^6 / z^3) - (10/3)x^6 y^4 z^3]

From here, it gets tricky, as a root can NOT be distributed over sums and differences

cubeRoot(a-b) is NOT the same as cubeRoot(a) - cubeRoot(b)

try to factor out common stuff.

[(27x^15 y^6 / z^3) - (10/3)x^6 y^4 z^3]

I would factor out 27 (yes), x^6, y^6 and z^3

to get:

(27x^6 y^6 z^3) [(x^9 / z^6) - (10/81)/y^2 ]

What do we have, so far?

cubeRoot(x^6y^4/3z^3) * cubeRoot(81x^9y^2 - 10z^6)

becomes

cubeRoot [(27x^15 y^6 / z^3) - (10/3)x^6 y^4 z^3]

becomes

cubeRoot {(27x^6 y^6 z^3) [(x^9 / z^6) - (10/81)/y^2 ]}

becomes

cubeRoot(27x^6 y^6 z^3) * cubeRoot[(x^9 / z^6) - (10/81)/y^2 ]

solve the first root:

3x^2y^2z * cubeRoot[(x^9 / z^6) - (10/81)/y^2 ]

The next step is to factor the remaining part of the root. In such problems, you normally get a "difference of powers" where there are tricks to factor. But because I do not know the original power of y, I would only be guessing.

The question has not been written properly, and apparently carries some mistakes.

First ensure the accuracy!