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 $\sqrt{x\sqrt[3]{x^2}}$
$$\sqrt{x\sqrt[3]{x^2}}=x^{\frac{1}{2}}\cdot (x^{\frac{2}{3}})^{\frac{1}{2}}=x^{\frac{1}{2}+\frac{2}{3}\cdot \frac{1}{2}}=x^{\frac{5}{6}}=\sqrt[6]{x^{5}}$$Wanna check your skills?
A quiz is based on this article and has 5 questions.
×
Quiz
2021-08-02
All categories
New content
- Circle: arc, sector and segment (Geometry)
- Circle: radius, diameter, circumference and area (Geometry)
- The law of cosines and examples (Geometry)
- The law of sines and examples (Geometry)
- Rewriting radicals into exponential form (Algebra)
- Infinite geometric sequence sum formula and examples (Algebra)
- Geometric sequence formulas and examples (Algebra)
- Arithmetic sequence formulas and examples (Algebra)