Question 450872
Solved Example using 3 of Exponent Rules



Simplify {{{(x^(1/3) - x^(-1/3))(x^(2/3) + 1 + x^(-2/3))}}}



Let {{{A = (x^(1/3) - x^(-1/3))(x^(2/3) + 1 + x^(-2/3))}}}



We know {{{x^(2/3) = x^(2 * 1/3) = x^((1/3) * 2 )= (x^(1/3))^2}}}



Similarly, {{{x^(-2/3) = (x^(-1/3))^2}}}



If we denote {{{x^(1/3)}}} by {{{a}}}, and {{{x^(-1/3 )}}}by {{{b}}},



then {{{ab = x^(1/3) * x^(-1/3) = 1}}} [Since pn and p-n are reciprocals to one another.]



Thus {{{A}}} becomes {{{(a - b)(a^2 + ab + b^2)}}}.



We know {{{(a - b)(a^2 + ab + b^2) = a^3 - b^3}}}



so, {{{ A = a^3 - b^3 = (x^(1/3))^3 - (x^(-1/3))^3}}}




= {{{x^((1/3 )* 3) - x^((-1/3) * 3) = x^1 - x^(-1) = x - 1/x}}}....this is your answer