Math Doubts

Power Rule of Exponents

Formula

${(b^{\displaystyle \, m})}^{\displaystyle n} \,=\, b^{\displaystyle \, mn}$

The power of an exponent with a base is equal to the product of the exponents with same base, is called the power rule of exponents.

Introduction

$b$, $m$ and $n$ are constants. The literals $b$ and $m$ formed an exponential term $b^{\displaystyle \,m}$. The literal $n$ is a power for exponential term $b^{\displaystyle \,m}$ and formed a special term ${(b^{\displaystyle \, m})}^{\displaystyle n}$ in exponential form.

The power $n$ of an exponent $m$ with base $b$ is equal to the product of the exponents $m$ and $n$ with base $b$.

${(b^{\displaystyle \, m})}^{\displaystyle n} \,=\, b^{\displaystyle \, mn}$

It is called as the power rule for exponents and used as a formula when a quantity is power of an exponential term.

Proof

Learn how to derive the power rule of exponents in algebraic form with understandable process.

Verification

${(2^3)}^4$ is a term in which an exponential term $2^3$ has $4$ as its power..

$\implies$ ${(2^3)}^4 \,=\, {(2 \times 2 \times 2)}^4$

$\implies$ ${(2^3)}^4$ $\,=\,$ $(2 \times 2 \times 2)$ $\times$ $(2 \times 2 \times 2)$ $\times$ $(2 \times 2 \times 2)$ $\times$ $(2 \times 2 \times 2)$

$\implies$ ${(2^3)}^4$ $\,=\,$ $2 \times 2 \times 2$ $\times$ $2 \times 2 \times 2$ $\times$ $2 \times 2 \times 2$ $\times$ $2 \times 2 \times 2$

$\implies$ ${(2^3)}^4$ $\,=\,$ $2^{12}$

In this example, $3$ and $4$ are exponents in the left-hand side of the equation but $12$ is power in the right-hand side of the equation. The number $12$ can be factored as the product of the numbers $3$ and $4$.

$\,\,\, \therefore \,\,\,\,\,\,$ ${(2^3)}^4$ $\,=\,$ $2^{\, 3 \times 4}$

Therefore, it is verified that the power of an exponential term is equal to the product of the exponents with same base.

Math Doubts

A best free mathematics education website for students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

Learn how to solve the maths problems in different methods with understandable steps.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2021 Math Doubts, All Rights Reserved