Math Doubts

Radical Power Rule of Exponents


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

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.


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.


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

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

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

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

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

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.

Math Doubts
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more