5.1: Definición de a
- Page ID
- 112409
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Para cualquier número real\(a\) y un número positivo\(n\),\(a^n\) es la multiplicación repetida de\(a\) por sí mismos\(n\) tiempos.
\[a^n= a\cdot a \cdot a\cdot a\cdot a\cdot a\cdot a\cdot a \ldots \ldots \cdot a \nonumber \]
Notación:
\(a\)es la base,\(n\) es el exponente positivo.
\(a^n\)se lee como”\(a\) elevado al poder de\(n\).”
Identificar la base y el exponente en expresiones.
\(2^4\),\(x^5\),\(\left(\dfrac{3}{7}\right)^7\),\((-3)^3\)
Solución
| Expresión | Base | Exponente |
|---|---|---|
| \(2^4\) | 2 | 4 |
| \(x^5\) | \(x\) | 5 |
| \(\left(\dfrac{3}{7}\right)^7\) | \(\dfrac{3}{7}\) | 7 |
| \((-3)^3\) | -3 | 3 |
Identificar la base y exponente de lo siguiente.
| Expresión | Base | Exponente |
|---|---|---|
| \(7^9\) | ||
| \((-11)^6\) | ||
| \(a^b\) | ||
| \(\left(\dfrac{11}{12}\right)^5\) | ||
| \(12^3\) | ||
| \(\left(-\dfrac{7}{3}\right)^2\) | ||
| \(x^7\) | ||
| \((2.56)^4\) |
Evaluar expresiones de la forma\(a^n\)
Cuando la base y el exponente es un valor numérico es posible evaluar una expresión escrita con un exponente. Para encontrar el valor, utilice la definición y expanda la expresión. Una vez expandida, multiplica y el resultado es el valor numérico de la expresión.
Expanda las siguientes expresiones y evalúe si es posible.
\(3^4\),\(\left(\dfrac{3}{5}\right)^3\),\(x^7\),\((3.12)^2\),\((-5)^3\),\((-y)^6\)
Solución
| \(3^4\) | \(= 3\cdot 3\cdot 3\cdot 3 = 81\) |
| \(\left(\dfrac{3}{5}\right)^3\) | \(\dfrac{3 }{5} \cdot \dfrac{3}{ 5 }\cdot \dfrac{3 }{5} = \dfrac{27 }{125}\) |
| \(x^7\) |
\(x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\) Nota: No se puede evaluar ya que x es desconocido |
| \((3.12)^2\) | \((3.12)\cdot (3.12) = 9.734\) |
| \((-5)^3\) | \(−5 \cdot −5 \cdot −5 = −12\) |
| \((-y)^6\) |
\(−y \cdot −y \cdot −y \cdot −y \cdot −y \cdot −y = y^6\) Nota: y es desconocido |
Expanda las siguientes expresiones y evalúe si es posible.
- \(7^3\)
- \(\left(−\dfrac{ 2 }{3}\right)^4\)
- \((−x)^7\)
- \((7.14)^2\)
- \((−3)^9\)
- \((z)^5\)
- \(\left(− \dfrac{11 }{33 }\right)^2\)
- \(6^5\)
- \(\left(\dfrac{x}{ y}\right)^4\)
- \(a^{10}\)
- \(\left(\dfrac{2}{x}\right)^3\)