# Radical Power Rule of Exponents

## Formula

$b^{\large \frac{\Large 1}{\displaystyle \normalsize n}} \,=\, \sqrt[\displaystyle n]{b}$

The fractional exponent of a number is equal to the square root or height order root of the number. It is called the radical power rule of the exponents.

### Introduction

In some special cases, the numbers contain rational numbers like $\dfrac{1}{2}$, $\dfrac{1}{3}$, $\dfrac{1}{4}$, $\cdots$ $\dfrac{1}{n}$ as exponents to express the square root or higher order roots of the numbers. Actually, the fractions are represented by radical symbols like square root $(\sqrt{\,\,\,\,})$, cube root $(\sqrt[\displaystyle 3]{\,\,\,\,})$, fourth root $(\sqrt[\displaystyle 4]{\,\,\,\,})$, $\cdots$ $n$-th root $(\sqrt[\displaystyle n]{\,\,\,\,})$ respectively in mathematics.

#### Examples

$(1) \,\,\,\,\,\,$ $6^{\large \frac{1}{2}} \,=\, \sqrt{6}$

$(2) \,\,\,\,\,\,$ $2^{\large \frac{1}{3}} \,=\, \sqrt[\displaystyle 3]{2}$

$(3) \,\,\,\,\,\,$ $7^{\large \frac{1}{4}} \,=\, \sqrt[\displaystyle 4]{7}$

$(4) \,\,\,\,\,\,$ $11^{\large \frac{1}{5}} \,=\, \sqrt[\displaystyle 5]{11}$

$(5) \,\,\,\,\,\,$ $9^{\large \frac{1}{6}} \,=\, \sqrt[\displaystyle 6]{9}$

#### Algebraic form

$b$ and $n$ are two literals. The $n$-th root of the literal number $b$ is written as $b^{\large \frac{\Large 1}{\displaystyle \normalsize n}}$. It is also expressed in radical form as $\sqrt[\displaystyle n]{b}$.

$\therefore \,\,\,\,\,\,$ $b^{\large \frac{\Large 1}{\displaystyle \normalsize n}} \,=\, \sqrt[\displaystyle n]{b}$

This power rule of exponents is called as the radical or fractional power rule of exponents.

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