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Laws of exponents

A power $a^n$ is a shorthand for multiplying $a$ by itself $n$ times. When multiplying the same base the exponents are added, when dividing they are subtracted, and for a power of a power they are multiplied. The derivation below shows why, by counting the $a$'s.

Symbols

Where the rule comes from

Multiplication: why the exponents are added

1. $$a^{\textcolor{#1D5FC2}{3}}\cdot a^{\textcolor{#A64300}{2}} = (\textcolor{#1D5FC2}{a\cdot a\cdot a})\cdot(\textcolor{#A64300}{a\cdot a})$$

   A power is repeated multiplication: $a^3$ is three $a$'s multiplied together, and $a^2$ is two.

2. $$= \textcolor{#1D5FC2}{a\cdot a\cdot a}\cdot \textcolor{#A64300}{a\cdot a}$$

   Remove the brackets. Now there are five $a$'s multiplied together: three blue and two orange.

3. $$= a^{\textcolor{#1D5FC2}{3}+\textcolor{#A64300}{2}} = a^5$$

   The count is $3 + 2 = 5$. With $m$ and $n$ $a$'s there are $m + n$ altogether, so $a^m\cdot a^n = a^{m+n}$.

Division: why the exponents are subtracted

1. $$\frac{a^{\textcolor{#1D5FC2}{5}}}{a^{\textcolor{#A64300}{2}}} = \frac{\textcolor{#1D5FC2}{a\cdot a\cdot a\cdot a\cdot a}}{\textcolor{#A64300}{a\cdot a}}$$

   Write the powers as multiplication: five $a$'s above the fraction bar and two below.

2. $$= \textcolor{#1D5FC2}{a\cdot a\cdot a} = a^{\textcolor{#1D5FC2}{5}-\textcolor{#A64300}{2}} = a^3$$

   Two $a$'s above and two below cancel, because $\frac{a}{a} = 1$. That leaves $5 - 2 = 3$ $a$'s.

Power of a power: why the exponents are multiplied

1. $$(a^{\textcolor{#1D5FC2}{2}})^{\textcolor{#A64300}{3}} = a^{\textcolor{#1D5FC2}{2}}\cdot a^{\textcolor{#1D5FC2}{2}}\cdot a^{\textcolor{#1D5FC2}{2}}$$

   Raising to the third power means multiplying by itself three times.

2. $$= a^{\textcolor{#1D5FC2}{2}+\textcolor{#1D5FC2}{2}+\textcolor{#1D5FC2}{2}} = a^{\textcolor{#1D5FC2}{2}\cdot \textcolor{#A64300}{3}} = a^6$$

   Use the multiplication rule and add the exponents: three twos are $2\cdot 3 = 6$.

Example

$2^{\textcolor{#1D5FC2}{3}}\cdot 2^{\textcolor{#A64300}{4}} = 2^{\textcolor{#1D5FC2}{3}+\textcolor{#A64300}{4}} = 2^7 = 128$.

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Part of Foundations of Mathematics: Powers, roots and scientific notation.