Math Doubts

Proof of Rational Power Rule of Exponents

The rational power rule of exponents can be derived in algebraic form to use it as a formula in mathematics.

Term with Rational Number as Power

$b$, $m$ and $n$ are three literals and they represent three different quantities. Take $b$ as a base and a fraction $\dfrac{m}{n}$ as exponent to form a special exponential term.

$b^{\Large \frac{m}{n}}$

Fractional Power in Product form

Now, write the fraction as product of two numbers to simplify the exponent.

$\implies$ $b^{\Large \frac{m}{n}} \,=\, b^{\, m \times \Large \frac{1}{n}}$

Simplifying the Expression

Now, use power rule of exponents to express the product of exponents as power of an exponent.

$\implies$ $b^{\Large \frac{m}{n}} \,=\, {\Big(b^m\Big)}^{\Large \frac{1}{n}} $

According to Radical power rule of exponents, the power of exponential term $b^m$ is a radical and it can be denoted by a radical symbol.

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

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved