5.3: La regla del cociente de los exponentes
- Page ID
- 112429
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Para cualquier número real\(a\) y números positivos\(m\) y\(n\), dónde\(m > n\).
La regla del cociente para los exponentes es la siguiente.
\(\dfrac{a^m }{a^n} = a^{ m−n}\)
Nota: Las bases DEBEN ser las mismas. El resultado tendrá la misma base.
Idea:
A partir de la última sección,
\(x^3 = \textcolor{blue}{x \cdot x \cdot x} \qquad x^5 = \textcolor{red}{x \cdot x \cdot x \cdot x \cdot x}\)
Su cociente
\(\dfrac{x^ 5 }{x^3} = \dfrac{\textcolor{red}{x \cdot x \cdot x \cdot x \cdot x }}{\textcolor{blue}{x \cdot x \cdot x }}= \dfrac{\textcolor{red}{\cancel{x \cdot x\cdot x \cdot x }\cdot x }}{\textcolor{blue}{\cancel{x \cdot x\cdot x }}}= \dfrac{\textcolor{red}{x \cdot x }}{1} = \textcolor{red}{x \cdot x}\).
Entonces,\(\dfrac{x^5 }{x^3 }= x^{5−3 }= x^2\)
Usar la regla de cociente de exponentes para simplificar expresiones.
- \(\dfrac{k^3 }{k^2}\)
- \(\dfrac{r^{32} }{r^{21}}\)
- \(\dfrac{\sqrt{2}^ 7 }{\sqrt{2 }^4}\)
- \(\dfrac{(−7)^9 }{(−7)^6}\)
- \(\dfrac{(x \sqrt{5})^8 }{x\sqrt{ 5}}\)
- \(\dfrac{(xy)^{18} }{(xy)^{17}}\)
Solución
| Expresión | Regla del cociente | Base |
| \(\dfrac{k^3 }{k^2}\) | \(k^{3−2 }= k\) | \(k\) |
| \(\dfrac{r^{32} }{r^{21}}\) | \(r^{32−21 }= r^{11}\) | \(r\) |
| \(\dfrac{\sqrt{2}^ 7 }{\sqrt{2 }^4}\) | \(\sqrt{2 }^{7−4 }= \sqrt{2 }^3\) | \(\sqrt{2}\) |
| \(\dfrac{(−7)^9 }{(−7)^6}\) | \((−7)^{9−6 }= (−7)^3\) | \(-7\) |
| \(\dfrac{(x \sqrt{5})^8 }{x\sqrt{ 5}}\) | \((x \sqrt{5})^{8−1 }= (x \sqrt{5})^7\) | \(x\sqrt{5}\) |
| \(\dfrac{(xy)^{18} }{(xy)^{17}}\) | \((xy)^{18−17 }= xy\) | \(xy\) |
Nota: En esta sección el exponente del numerador fue mayor que el exponente del denominador. Ese no siempre será el caso. El caso donde el exponente en el denominador es mayor que el exponente en el numerador se discutirá en una sección posterior.
Utilice la regla de cociente de exponentes para simplificar la expresión dada.
- \(\dfrac{−y ^{13} }{−y^7}\)
- \(\dfrac{(2x)^{25}}{ 2x}\)
- \(\dfrac{\sqrt{7 }^{17 }}{\sqrt{7 }^{12}}\)
- \(\dfrac{(−7)^9 }{(−7)^6}\)
- \(\dfrac{(x + y) ^{78}}{ (x + y)^{43}}\)
- \(\dfrac{\sqrt{xy }^{15 }}{\sqrt{xy }^{11}}\)