I think you want to show that it is (a-b), no?
since (x^3 - y^3) = (x-y)(x^2+xy+y^2)
just substitute x=a^1/3 and y=b^1/3
Or, just multiply it out longhand.
Or, finish asking the question.
Show that ( a1/3 - b1/3) ( a2/3 +a1/3 * b1/3 + b2/3 )?
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