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

## Exercise

Simplify the expression:

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

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

## 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:

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

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