Powers and Roots – Simplify an expression with roots – Exercise 5682

Exercise

Simplify the expression:

\frac{5}{\sqrt[3]{3}+\sqrt[3]{2}}

Final Answer

Show final answer

Solution

Using Powers and Roots rules we get:

\frac{5}{\sqrt[3]{3}+\sqrt[3]{2}}=

We get rid of the roots in the denominator:

=\frac{5(3^{\frac{2}{3}}-3^{\frac{1}{3}}\cdot 2^{\frac{1}{3}}+2^{\frac{2}{3}})}{(\sqrt[3]{3}+\sqrt[3]{2})(3^{\frac{2}{3}}-3^{\frac{1}{3}}\cdot 2^{\frac{1}{3}}+2^{\frac{2}{3}})}=

=\frac{5(3^{\frac{2}{3}}-3^{\frac{1}{3}}\cdot 2^{\frac{1}{3}}+2^{\frac{2}{3}})}{3+2}=

=3^{\frac{2}{3}}-3^{\frac{1}{3}}\cdot 2^{\frac{1}{3}}+2^{\frac{2}{3}}=

One can further simplify the result:

=\sqrt[3]{9}-\sqrt[3]{6}+\sqrt[3]{4}