5.8: Poder de una regla de cociente para exponentes
- Page ID
- 112439
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)El poder de una regla de cociente para exponentes se centrará en lo que le sucede a un cociente cuando se eleva a algún poder.
Para cualquier número real\(a\) y\(b\) y cualquier entero\(n\), la potencia de una regla de cociente para exponentes es la siguiente:
\(\left( \dfrac{a }{b} \right)^n = \dfrac{a^n }{b^n }\),
donde\(b \neq 0\).
Simplifique lo siguiente usando la regla de potencia de un cociente para exponentes.
Simplifique lo siguiente usando la regla de potencia de un cociente para exponentes.
\(\left( \dfrac{a }{b} \right)^4\)
Solución
\(\begin{aligned} &\left( \dfrac{a}{ b} \right)^4 && \text{Given} \\ &= \dfrac{a }{b} \cdot \dfrac{a }{b} \cdot \dfrac{a }{b} \cdot \dfrac{a }{b} &&\text{Expand using the exponent definition} \\ &= \dfrac{a^4 }{b^4} && \text{Multiply as needed to simplify} \end{aligned}\)
\(\left(\dfrac{x^2 }{3y^5} \right)^3\)
Solución
\(\begin{aligned} &\left( \dfrac{x^2 }{3y^5 }\right)^3 && \text{Given} \\ &= \dfrac{x^{2\cdot 3 }}{3^3 \cdot y^{5\cdot 3 }} && \text{power of quotient rule for exponents applied} \\ &= \dfrac{x^6 }{3^3 \cdot y^{15 }} &&\text{Simplify exponent product} \\ &= \dfrac{x^6 }{27y^{15 }} && \text{Multiply as needed to simplify.} \end{aligned}\)
\(\left( \dfrac{2x }{y }\right)^{−3}\)
Solución
\(\begin{aligned} &\left( \dfrac{2x }{y }\right)^{−3 } &&\text{Given} \\ &= \left( \dfrac{y }{2x} \right)^3 && \text{Negative exponent rule applied} \\ &= \dfrac{y^3 }{2^3 \cdot x^3} && \text{Power of a quotient rule for exponents applied.} \\ &= \dfrac{y^3 }{8x^3 } && \text{Multiply as needed to simplify.} \end{aligned}\)
El orden en que se apliquen las reglas de exponentes no importa. En el ejemplo tres, los pasos 2 y 3 se pueden hacer en cualquier orden. Los resultados serán los mismos.
Simplifica la expresión usando el poder de una regla de cociente para exponentes.
- \(\left( \dfrac{p^4 }{p^7 }\right) ^3\)
- \(−\left(\dfrac{ x^2 \cdot x^3 }{x \cdot y^3} \right) ^2\)
- \(\left( \dfrac{5x^3 }{2y^{13 }}\right) ^{−2}\)
- \(\left( \dfrac{2c^3}{ c^4} \right) ^3\)
- \(\left( \dfrac{a ^{−7}b }{a^2b^{−4 }}\right)^3\)
- \(\left( \dfrac{f^{−7 }}{f^5 }\right)^9\)
- \(\left(\dfrac{ xy^2z^3}{ x^3y^2z} \right) ^5\)