Question

Express with rational denominator $\frac{4}{\sqrt[3]{39}-\sqrt[3]{33}+1}$.

Answer 1

The expression

$\frac{4}{\sqrt[3]{39}-\sqrt[3]{33}+1}$.

$=\frac{4}{3^{\frac{2}{3}}-3^{\frac{1}{3}}+1}$

$=\frac{4(3^{\frac{1}{3}}+1)}{(3^{\frac{1}{3}}+1)(3^{\frac{2}{3}}-3^{\frac{1}{3}}+1)}$

$=\frac{4(3^{\frac{1}{3}}+1)}{3+1}=3^{\frac{1}{3}}+1$