8.4E: Ejercicios
- Page ID
- 112567
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)La práctica hace la perfección
En los siguientes ejercicios, escribe como expresión radical.
- a.\(x^{\frac{1}{2}}\) b.\(y^{\frac{1}{3}}\) c.\(z^{\frac{1}{4}}\)
- a.\(r^{\frac{1}{2}}\) b.\(s^{\frac{1}{3}}\) c.\(t^{\frac{1}{4}}\)
- a.\(u^{\frac{1}{5}}\) b.\(v^{\frac{1}{9}}\) c.\(w^{\frac{1}{20}}\)
- a.\(g^{\frac{1}{7}}\) b.\(h^{\frac{1}{5}}\) c.\(j^{\frac{1}{25}}\)
- Contestar
-
1. a.\(\sqrt{x}\) b.\(\sqrt[3]{y}\) c.\(\sqrt[4]{z}\)
3. a.\(\sqrt[5]{u}\) b.\(\sqrt[9]{v}\) c.\(\sqrt[20]{w}\)
En los siguientes ejercicios, escribe con un exponente racional.
- a.\(\sqrt[7]{x}\) b.\(\sqrt[9]{y}\) c.\(\sqrt[5]{f}\)
- a.\(\sqrt[8]{4}\) b.\(\sqrt[10]{s}\) c.\(\sqrt[4]{t}\)
- a.\(\sqrt[3]{7c}\) b.\(\sqrt[7]{12d}\) c.\(2\sqrt[4]{6b}\)
- a.\(\sqrt[4]{5x}\) b.\(\sqrt[8]{9y}\) c.\(7\sqrt[5]{3z}\)
- a.\(\sqrt{21p}\) b.\(\sqrt[4]{8q}\) c.\(4\sqrt[6]{36r}\)
- a.\(\sqrt[3]{25a}\) b.\(\sqrt{3b}\) c.\(\sqrt[8]{40c}\)
- Contestar
-
1. a.\(x^{\frac{1}{7}}\) b.\(y^{\frac{1}{9}}\) c.\(f^{\frac{1}{5}}\)
3. a.\((7 c)^{\frac{1}{4}}\) b.\((12 d)^{\frac{1}{7}}\) c.\(2(6 b)^{\frac{1}{4}}\)
5. a.\((21 p)^{\frac{1}{2}}\) b.\((8 q)^{\frac{1}{4}}\) c.\(4(36 r)^{\frac{1}{6}}\)
En los siguientes ejercicios, simplifique.
- a.\(81^{\frac{1}{2}}\) b.\(125^{\frac{1}{3}}\) c.\(64^{\frac{1}{2}}\)
- a.\(625^{\frac{1}{4}}\) b.\(243^{\frac{1}{5}}\) c.\(32^{\frac{1}{5}}\)
- a.\(16^{\frac{1}{4}}\) b.\(16^{\frac{1}{2}}\) c.\(625^{\frac{1}{4}}\)
- a.\(64^{\frac{1}{3}}\) b.\(32^{\frac{1}{5}}\) c.\(81^{\frac{1}{4}}\)
- a.\((-216)^{\frac{1}{3}}\) b.\(-216^{\frac{1}{3}}\) c.\((216)^{-\frac{1}{3}}\)
- a.\((-1000)^{\frac{1}{3}}\) b.\(-1000^{\frac{1}{3}}\) c.\((1000)^{-\frac{1}{3}}\)
- a.\((-81)^{\frac{1}{4}}\) b.\(-81^{\frac{1}{4}}\) c.\((81)^{-\frac{1}{4}}\)
- a.\((-49)^{\frac{1}{2}}\) b.\(-49^{\frac{1}{2}}\) c.\((49)^{-\frac{1}{2}}\)
- a.\((-36)^{\frac{1}{2}}\) b.\(-36^{\frac{1}{2}}\) c.\((36)^{-\frac{1}{2}}\)
- a.\((-16)^{\frac{1}{4}}\) b.\(-16^{\frac{1}{4}}\) c.\(16^{-\frac{1}{4}}\)
- a.\((-100)^{\frac{1}{2}}\) b.\(-100^{\frac{1}{2}}\) c.\((100)^{-\frac{1}{2}}\)
- a.\((-32)^{\frac{1}{5}}\) b.\((243)^{-\frac{1}{5}}\) c.\(-125^{\frac{1}{3}}\)
- Contestar
-
1. a.\(9\) b.\(5\) c.\(8\)
3. a.\(2\) b.\(4\) c.\(5\)
5. a.\(-6\) b.\(-6\) c.\(\frac{1}{6}\)
7. a. no real b.\(-3\) c.\(\frac{1}{3}\)
9. a. no real b.\(-6\) c.\(\frac{1}{6}\)
11. a. no real b.\(-10\) c.\(\frac{1}{10}\)
En los siguientes ejercicios, escribe con un exponente racional.
- a.\(\sqrt{m^{5}}\) b.\((\sqrt[3]{3 y})^{7}\) c.\(\sqrt[5]{\left(\dfrac{4 x}{5 y}\right)^{3}}\)
- a.\(\sqrt[4]{r^{7}}\) b.\((\sqrt[5]{2 p q})^{3}\) c.\(\sqrt[4]{\left(\dfrac{12 m}{7 n}\right)^{3}}\)
- a.\(\sqrt[5]{u^{2}}\) b.\((\sqrt[3]{6 x})^{5}\) c.\(\sqrt[4]{\left(\dfrac{18 a}{5 b}\right)^{7}}\)
- a.\(\sqrt[3]{a}\) b.\((\sqrt[4]{21 v})^{3}\) c.\(\sqrt[4]{\left(\dfrac{2 x y}{5 z}\right)^{2}}\)
- Contestar
-
1. a.\(m^{\frac{5}{2}}\) b.\((3 y)^{\frac{7}{3}}\) c.\(\left(\dfrac{4 x}{5 y}\right)^{\frac{3}{5}}\)
3. a.\(u^{\frac{2}{5}}\) b.\((6 x)^{\frac{5}{3}}\) c.\(\left(\dfrac{18 a}{5 b}\right)^{\frac{7}{4}}\)
En los siguientes ejercicios, simplifique.
- a.\(64^{\frac{5}{2}}\) b.\(81^{\frac{-3}{2}}\) c.\((-27)^{\frac{2}{3}}\)
- a.\(25^{\frac{3}{2}}\) b.\(9^{-\frac{3}{2}}\) c.\((-64)^{\frac{2}{3}}\)
- a.\(32^{\frac{2}{5}}\) b.\(27^{-\frac{2}{3}}\) c.\((-25)^{\frac{1}{2}}\)
- a.\(100^{\frac{3}{2}}\) b.\(49^{-\frac{5}{2}}\) c.\((-100)^{\frac{3}{2}}\)
- a.\(-9^{\frac{3}{2}}\) b.\(-9^{-\frac{3}{2}}\) c.\((-9)^{\frac{3}{2}}\)
- a.\(-64^{\frac{3}{2}}\) b.\(-64^{-\frac{3}{2}}\) c.\((-64)^{\frac{3}{2}}\)
- Contestar
-
1. a.\(32,768\) b.\(\frac{1}{729}\) c.\(9\)
3. a.\(4\) b.\(\frac{1}{9}\) c. no real
5. a.\(-27\) b.\(-\frac{1}{27}\) c. no real
En los siguientes ejercicios, simplifique. Supongamos que todas las variables son positivas.
- a.\(c^{\frac{1}{4}} \cdot c^{\frac{5}{8}}\) b.\(\left(p^{12}\right)^{\frac{3}{4}}\) c.\(\dfrac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}\)
- a.\(6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}\) b.\(\left(b^{15}\right)^{\frac{3}{5}}\) c.\(\dfrac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}\)
- a.\(y^{\frac{1}{2}} \cdot y^{\frac{3}{4}}\) b.\(\left(x^{12}\right)^{\frac{2}{3}}\) c.\(\dfrac{m^{\frac{5}{8}}}{m^{\frac{13}{8}}}\)
- a.\(q^{\frac{2}{3}} \cdot q^{\frac{5}{6}}\) b.\(\left(h^{6}\right)^{\frac{4}{3}}\) c.\(\dfrac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}\)
- a.\(\left(27 q^{\frac{3}{2}}\right)^{\frac{4}{3}}\) b.\(\left(a^{\frac{1}{3}} b^{\frac{2}{3}}\right)^{\frac{3}{2}}\)
- a.\(\left(64 s^{\frac{3}{7}}\right)^{\frac{1}{6}}\) b.\(\left(m^{\frac{4}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{4}}\)
- a.\(\left(16 u^{\frac{1}{3}}\right)^{\frac{3}{4}}\) b.\(\left(4 p^{\frac{1}{3}} q^{\frac{1}{2}}\right)^{\frac{3}{2}}\)
- a.\(\left(625 n^{\frac{8}{3}}\right)^{\frac{3}{4}}\) b.\(\left(9 x^{\frac{2}{5}} y^{\frac{3}{5}}\right)^{\frac{5}{2}}\)
- a.\(\dfrac{r^{\frac{5}{2}} \cdot r^{-\frac{1}{2}}}{r^{-\frac{3}{2}}}\) b.\(\left(\dfrac{36 s^{\frac{1}{5}} t^{-\frac{3}{2}}}{s^{-\frac{9}{5}} t^{\frac{1}{2}}}\right)^{\frac{1}{2}}\)
- a.\(\dfrac{a^{\frac{3}{4}} \cdot a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}}\) b.\(\left(\dfrac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}}\right)^{\frac{1}{3}}\)
- a.\(\dfrac{c^{\frac{5}{3}} \cdot c^{-\frac{1}{3}}}{c^{-\frac{2}{3}}}\) b.\(\left(\dfrac{8 x^{\frac{5}{3}} y^{-\frac{1}{2}}}{27 x^{-\frac{4}{3}} y^{\frac{5}{2}}}\right)^{\frac{1}{3}}\)
- a.\(\dfrac{m^{\frac{7}{4}} \cdot m^{-\frac{5}{4}}}{m^{-\frac{2}{4}}}\) b.\(\left(\dfrac{16 m^{\frac{1}{5}} n^{\frac{3}{2}}}{81 m^{\frac{9}{5}} n^{-\frac{1}{2}}}\right)^{\frac{1}{4}}\)
- Contestar
-
1. a.\(c^{\frac{7}{8}}\) b.\(p^{9}\) c.\(\frac{1}{r}\)
3. a.\(y^{\frac{5}{4}}\) b.\(x^{8}\) c.\(\dfrac{1}{m}\)
5. a.\(81 q^{2}\) b.\(a^{\frac{1}{2}} b\)
7. a.\(8 u^{\frac{1}{4}}\) b.\(8 p^{\frac{1}{2}} q^{\frac{3}{4}}\)
9. a.\(r^{\frac{7}{2}}\) b.\(\dfrac{6 s}{t}\)
11. a.\(c^{2}\) b.\(\dfrac{2x}{3y}\)
- Mostrar dos métodos algebraicos diferentes para simplificar\(4^{\frac{3}{2}}\). Explica todos tus pasos.
- Explique por qué la expresión\((-16)^{\frac{3}{2}}\) no puede ser evaluada.
- Contestar
-
1. Las respuestas variarán.
Autocomprobación
a. después de completar los ejercicios, utilice esta lista de verificación para evaluar su dominio de los objetivos de esta sección.
b. ¿Qué te dice esta lista de verificación sobre tu dominio de esta sección? ¿Qué pasos tomarás para mejorar?