According to my textbook it should be done like you mentioned by cubing both members.
I am not aware of any legitimate solution method which would involve cubing two of the terms. One can only cube both entire sides.
I do not see the sense of the rest of what the text appears to say. Moving the 5 to the right-hand side and cubing both sides gives:
I do not see how your text then derives an equation with only one cube root on the left-hand side.