Math Doubts

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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