The answer is (9+z^6)^(2/3) /4

You made a mistake. Your substitution is good, but you went astray. After your substitution, simply treat the entity as a polynomial. Raise the exponent by one and divide by the new exponent. Then, resubstitute the original expression back in. Lastly, the sure-fire proof is to differentiate your answer and see that you get the original expression.