5.1: Definition of aⁿ
- Page ID
- 45181
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For any real number \(a\) and a positive number \(n\), \(a^n\) is the repeated multiplication of \(a\) by itself \(n\) times.
\[a^n= a\cdot a \cdot a\cdot a\cdot a\cdot a\cdot a\cdot a \ldots \ldots \cdot a \nonumber \]
Notation:
\(a\) is the base, \(n\) is the positive exponent.
\(a^n\) is read as ”\(a\) raised to the power of \(n\).”
Identifying the base and exponent in expressions.
\(2^4\), \(x^5\), \(\left(\dfrac{3}{7}\right)^7\), \((-3)^3\)
Solution
| Expression | Base | Exponent |
|---|---|---|
| \(2^4\) | 2 | 4 |
| \(x^5\) | \(x\) | 5 |
| \(\left(\dfrac{3}{7}\right)^7\) | \(\dfrac{3}{7}\) | 7 |
| \((-3)^3\) | -3 | 3 |
Identify the base and exponent of the following.
| Expression | Base | Exponent |
|---|---|---|
| \(7^9\) | ||
| \((-11)^6\) | ||
| \(a^b\) | ||
| \(\left(\dfrac{11}{12}\right)^5\) | ||
| \(12^3\) | ||
| \(\left(-\dfrac{7}{3}\right)^2\) | ||
| \(x^7\) | ||
| \((2.56)^4\) |
Evaluating expressions of the form \(a^n\)
When the base and exponent is a numerical value it is possible to evaluate an expression written in with an exponent. To find the value, use the definition and expand the expression. Once expanded, multiply and the result is the numerical value of the expression.
Expand the following expressions and evaluate if possible.
\(3^4\), \(\left(\dfrac{3}{5}\right)^3\), \(x^7\), \((3.12)^2\), \((-5)^3\), \((-y)^6\)
Solution
| \(3^4\) | \(= 3\cdot 3\cdot 3\cdot 3 = 81\) |
| \(\left(\dfrac{3}{5}\right)^3\) | \(\dfrac{3 }{5} \cdot \dfrac{3}{ 5 }\cdot \dfrac{3 }{5} = \dfrac{27 }{125}\) |
| \(x^7\) |
\(x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\) Note: Can’t evaluate since x is unknown |
| \((3.12)^2\) | \((3.12)\cdot (3.12) = 9.734\) |
| \((-5)^3\) | \(−5 \cdot −5 \cdot −5 = −12\) |
| \((-y)^6\) |
\(−y \cdot −y \cdot −y \cdot −y \cdot −y \cdot −y = y^6\) Note: y is unknown |
Expand the following expressions and evaluate if possible.
- \(7^3\)
- \(\left(−\dfrac{ 2 }{3}\right)^4\)
- \((−x)^7\)
- \((7.14)^2\)
- \((−3)^9\)
- \((z)^5\)
- \(\left(− \dfrac{11 }{33 }\right)^2\)
- \(6^5\)
- \(\left(\dfrac{x}{ y}\right)^4\)
- \(a^{10}\)
- \(\left(\dfrac{2}{x}\right)^3\)