Skip to main content Contents Prev Up Next \(\newcommand{\R}{\mathbb R}
\newcommand{\reference}[1]{\overline{#1}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 2.1 Negative Exponents \(x^{-n}\)
A positive integer exponent indicates repeated multiplication. For instance,
\begin{equation*}
2^4 = \underbrace{2\cdot 2\cdot 2\cdot 2}_{\text{4 factors of 2}} = 16.
\end{equation*}
Definition 2.1 . Positive Integer Exponents.
Suppose \(n = 1,2,3,4,\ldots\) is a positive integer. We define
\begin{equation*}
a^n = \underbrace{a \cdot a \cdot \cdot \cdots \cdot a}_{\text{$n$ factors of $a$}}.
\end{equation*}
In particular, \(a^1 = a\text{.}\)
When multiplying powers with the same base, you should add the exponents. For example,
\begin{equation*}
2^3 \cdot 2^4 = \underbrace{2\cdot 2\cdot 2}_{\text{3 factors}} \cdot \underbrace{2\cdot 2\cdot 2\cdot 2}_{\text{4 factors}} = 2^{3+4} = 2^7.
\end{equation*}
Principle 2.2 . Multiplying Powers with Same Base.
\begin{equation*}
\boxed{a^n\cdot a^m = a^{n+m}.}
\end{equation*}
What would happen if one of the exponents was zero? For example, according to our law of exponents above,
\begin{equation*}
2^0\cdot 2^4 = 2^{0+4} = 2^4.
\end{equation*}
So that multiplying by \(2^0\) didn’t change the other factor; it’s as if we were multiplying by one! In fact, we will define \(2^0 = 1\text{.}\)
Definition 2.3 . Zero Exponent.
Suppose \(a\neq 0\) is a non-zero real number. We define
\begin{equation*}
a^0 = 1.
\end{equation*}
Note that we leave undefined the expression \(0^0\text{.}\)
When dividing powers of the same base, you should subtract the exponents. For example,
\begin{equation*}
\frac{2^6}{2^2} = \frac{2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2}{2\cdot 2} = 2^{6-2} =2^4.
\end{equation*}
Principle 2.4 . Dividing Powers with Same Base.
Suppose \(a\neq 0\) is a non-zero real number. Then
\begin{equation*}
\frac{a^n}{a^m} = a^{n-m}.
\end{equation*}
What would happen if one of the exponents was negative? It should be consistent with the above laws of exponents. Consider,
\begin{equation*}
2^{-1} \cdot 2^3 = 2^{-1+3} = 2^{2}.
\end{equation*}
Observe that multiplying by \(2^{-1}\) had the effect of cutting \(2^3\) in half. This suggests that \(2^{-1}\) is the same as the reciprocal of two, that is
\begin{equation*}
2^{-1} = \frac{1}{2}.
\end{equation*}
In general, we define the opposite exponent to mean the reciprocal.
Definition 2.5 . Negative Integer Exponents.
Suppose \(a\neq 0\) is a non-zero real number. We define
\begin{equation*}
a^{-n} = \frac{1}{a^n}
\end{equation*}
Example 2.6 .
Evaluate each of the following expressions without using a calculator. Be sure to apply the correct
order of operations 1.9 .
\(\displaystyle \displaystyle -2^4\)
\(\displaystyle \displaystyle (-2)^4\)
\(\displaystyle \displaystyle 2^{-4}\)
\(\displaystyle \displaystyle \frac{1}{2^{-4}}\)
\(\displaystyle \displaystyle \frac{1}{-2^{4}}\)
\(\displaystyle \displaystyle 1-2^{4}\)
Solution .
Apply the exponent before multiplying by
\(-1\)
\begin{equation*}
-2^4 = - 16.
\end{equation*}
This is an
even power of a negative number
\begin{equation*}
(-2)^4 = (-2)(-2)(-2)(-2) = + 16 = 16.
\end{equation*}
A negative exponent is equivalent to the reciprocal
\begin{equation*}
2^{-4} = \frac{1}{2^4} = \frac{1}{16}.
\end{equation*}
This applies for a power in the denominator as well
\begin{equation*}
\frac{1}{2^{-4}} = 2^4 = 16.
\end{equation*}
Apply the exponent before multiplying by
\(-1\) or dividing
\begin{equation*}
\frac{1}{-2^{4}} = \frac{1}{-16} = - \frac{1}{16}.
\end{equation*}
Apply the exponent before subtracting
\begin{equation*}
1-2^{4} = 1 - 16 = -15.
\end{equation*}
