# 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.

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