Basic Integration + Root + Exponential issue

Question

I have the following question : $\int ( 27e^{9x} + e^{12x} )^{1/3} dx $

However when I solved it I simplified it first to:

$\int ( 27e^{9x} + e^{12x} )^{1/3} dx = \int \sqrt[3]{27e^{9x} + e^{12x}} dx = \int 3e^{3x} + e^{4} dx $

But the solution doesn't mention anything about that :

Source : https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/exponsoldirectory/ExponSol3.html#SOLUTION 10

Answer 1

HINT:

In general $$\sqrt[3]{a^3+b^3}\ne a+b$$ as it will imply $$a^3+b^3=(a+b)^3\iff ab(a+b)=0$$