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:

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