1.4: Exponents
- Page ID
- 87276
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1.4.1 Learning Objectives
- Read, interpret, and write exponential notation
Exponential Notation
Definition: Exponential Notation
Just as multiplication is a description of repeated addition, exponential notation is a description of repeated multiplication.
Suppose we have the repeated multiplication \(8 \cdot 8 \cdot 8 \cdot 8 \cdot 8\).
The factor 8 is repeated 5 times.
Definition: Exponent
Exponential notation uses a superscript for the number of times the factor is repeated. The superscript is placed on the repeated factor, \(8^5\), in this case. The superscript is called an exponent. An exponent records the number of identical factors that are repeated in a multiplication.
If \(x\) is any real number and \(n\) is a natural number, then
\(x^{n}=\underbrace{x \cdot x \cdot x \cdot \ldots \cdot x}_{n \text { factors of } x}\)
An exponent records the number of identical factors in a multiplication.
Example 1
Write the following multiplications using exponents.
- \(3 \cdot 3\)
- \(62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62\)
Solution
- Since the factor 3 appears 2 times, we record this as \(3^2\).
- Since the factor 62 appears 9 times, we record this as \(62^9\).
Example 3
Expand (write without exponents) each number.
- \(12^4\)
- \(706^3\)
- \(15^1\)
Solution
- The exponent 4 is recording 4 factors of 12 in a multiplication. Thus, \(12^4 = 12 \cdot 12 \cdot 12 \cdot 12\).
- The exponent 3 is recording 3 factors of 706 in a multiplication. Thus, \(706^3 = 706 \cdot 706 \cdot 706\).
- The exponent 1 is recording 1 factor of 15 in a multiplication. Thus, \(15^1 = 15\).
Try It Now 1
Write the following using exponents.
\(37 \cdot 37\)
- Answer
-
\(37^2\)
Reading Exponential Notation
In \(x^n\),
\(x\) is the base
\(n\) is the exponent
The number represented by \(x^n\) is called a power
The term \(x^n\) is read as "\(x\) to the \(n\)th."
In a number such as \(8^5\).
Base
8 is called the base.
Exponent, Power
5 is called the exponent, or power. \(8^5\) is read as "eight to the fifth power," or more simply as "eight to the fifth," or "the fifth power of eight."
Squared
When a number is raised to the second power, it is said to be squared. The number \(5^2\) can be read as
5 to the second power, or
5 to the second, or
5 squared.
Cubed
When a number is raised to the third power, it is said to be cubed. The number \(5^3\) can be read as
5 to the third power, or
5 to the third, or
5 cubed.
When a number is raised to the power of 4 or higher, we simply say that that number is raised to that particular power. The number \(5^8\) can be read as
5 to the eighth power, or just
5 to the eighth.