Question

Solve : $(x^{\frac{1}{3}}+x^{\frac{-1}{3}})(x^{\frac{2}{3}}-1+x^{\frac{-2}{3}})$

Answer 1

$(x^{\frac{1}{3}}+x^{\frac{-1}{3}})(x^{\frac{2}{3}}-1+x^{\frac{-2}{3}})$

$=(x^{\frac{1}{3}}+x^{\frac{-1}{3}})[{\left(x^{\frac{1}{3}}\right)}^2-\left(x^{\frac{1}{3}}\right)\left(x^{\frac{-1}{3}}\right)+{\left(x^{\frac{-1}{3}}\right)}^2]$

Let $x^{\frac{1}{3}}=a$ and $x^{\frac{-1}{3}}=b$

Now the expression is equivalent to:

$(a+b)(a^2-ab+b^2)$$=a^3+b^3$

$⟹{\left(x^{\frac{1}{3}}\right)}^3+{\left(x^{\frac{-1}{3}}\right)}^3$$=x+x^{-1}$

$=x+\frac{1}{x}$