Home · Algebra · Rewriting radicals into exponential form

We can rewrite every radical as an exponent. Frequently this step helps easily to solve the task of complex radicals.

By using the following property we can rewrite radicals into exponential form and on the contrary.

$$a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$$

Solved examples

Rewrite radical $\sqrt[5]{x^3}$ into exponential form.

$$\sqrt[5]{x^3}=x^{\frac{3}{5}}$$

Simplify the expression $\sqrt{x}\cdot \sqrt[3]{x}: \sqrt[5]{x^2}$

$$\sqrt{x}\cdot \sqrt[3]{x}: \sqrt[5]{x^2}=x^{\frac{1}{2}}\cdot x^{\frac{1}{3}}: x^{\frac{2}{5}}=x^{\frac{1}{2}+\frac{1}{3}-\frac{2}{5}}=x^{\frac{13}{30}}=\sqrt[30]{x^{13}}$$

Simplify the expression $\sqrt{x\sqrt[3]{x^2}}$

$$\sqrt{x\sqrt[3]{x^2}}=x^{\frac{1}{2}}\cdot (x^{\frac{2}{3}})^{\frac{1}{2}}=x^{\frac{1}{2}+\frac{2}{3}\cdot \frac{1}{2}}=x^{\frac{5}{6}}=\sqrt[6]{x^{5}}$$