${(b^{\displaystyle \, m})}^{\displaystyle n} \,=\, b^{\displaystyle \, mn}$
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.
$b$, $m$ 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.
Learn how to derive the power rule of exponents in algebraic form with understandable process.
${(2^3)}^4$ 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.
A best free mathematics education website for students, teachers and researchers.
Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.
Learn how to solve the maths problems in different methods with understandable steps.
Copyright © 2012 - 2022 Math Doubts, All Rights Reserved