Ejercicio 221 – Solucionario Baldor

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DESCARGA

CAPITULO XXX

TEORIA DE LOS EXPONENTES

Ejercicio 221

Expresar con signo radical y exponentes positivos:

$\begin{matrix} x^{- \frac{1}{2}} & = \frac{1}{x^{\frac{1}{2}}} \\ = \frac{1}{\sqrt{x}} \end{matrix}$
$\begin{matrix} \frac{1}{a^{- \frac{1}{2}} b^{\frac{2}{3}}} & = \frac{a^{\frac{1}{2}}}{b^{\frac{2}{3}}} \\ = \frac{\sqrt{a}}{\sqrt[3]{b^{2}}} \end{matrix}$
$\begin{matrix} 5 a^{\frac{5}{7}} b^{- \frac{1}{3}} & = \frac{5 a^{\frac{5}{7}}}{b^{\frac{1}{3}}} \\ = \frac{5 \sqrt[7]{a^{5}}}{\sqrt[3]{b}} \end{matrix}$
$\begin{matrix} \frac{3 x^{- 1}}{x^{- \frac{1}{2}}} & = 3 x^{- 1} x^{\frac{1}{2}} \\ = 3 x^{- 1 + \frac{1}{2}} \\ = 3 x^{- \frac{1}{2}} \\ = \frac{3}{x^{\frac{1}{2}}} \\ = \frac{3}{\sqrt{x}} \end{matrix}$
$\begin{matrix} 2 m^{- \frac{2}{5}} n^{\frac{3}{4}} & = \frac{2 n^{\frac{3}{4}}}{m^{\frac{2}{5}}} \\ = \frac{2 \sqrt[4]{n^{3}}}{\sqrt[5]{m^{2}}} \end{matrix}$
$\frac{1}{4 x^{\frac{1}{3}}} = \frac{1}{4 \sqrt[3]{x}}$
$\begin{matrix} \frac{x^{\frac{3}{5}}}{y^{- \frac{2}{3}}} & = x^{\frac{3}{5}} y^{\frac{2}{3}} \\ = \sqrt[5]{x^{3}} \sqrt[3]{y^{2}} \end{matrix}$
$\begin{matrix} \frac{3 a^{- \frac{3}{2}}}{x^{\frac{1}{4}}} & = \frac{3}{a^{\frac{3}{2}} x^{\frac{1}{4}}} \\ = \frac{3}{\sqrt[2]{a^{3}} \sqrt[4]{x}} \\ = \frac{3}{\sqrt[2]{a^{2} . a} \sqrt[4]{x}} \\ = \frac{3}{a \sqrt[2]{a} \sqrt[4]{x}} \end{matrix}$
$\begin{matrix} \frac{a^{- \frac{1}{2}}}{4 a^{2}} & = \frac{1}{4 a^{2} a^{\frac{1}{2}}} \\ = \frac{1}{4 a^{\frac{5}{2}}} \\ = \frac{1}{4 \sqrt{a^{5}}} \\ = \frac{1}{4 \sqrt{a^{2} . a^{3}}} \\ = \frac{1}{4 a \sqrt{a^{3}}} \end{matrix}$
$\begin{matrix} x^{- \frac{2}{3}} y^{\frac{3}{5}} z^{- \frac{4}{7}} & = \frac{y^{\frac{3}{5}}}{x^{\frac{2}{3}} z^{\frac{4}{7}}} \\ = \frac{\sqrt[5]{y^{3}}}{\sqrt[3]{x^{2}} \sqrt[7]{z^{4}}} \end{matrix}$
$\begin{matrix} x^{- 2} m^{- 3} n^{- \frac{2}{5}} & = \frac{1}{x^{2} m^{3} n^{\frac{2}{5}}} \\ = \frac{1}{x^{2} m^{3} \sqrt[5]{n^{2}}} \end{matrix}$
$\begin{matrix} {(a^{- \frac{1}{2}})}^{3} & = a^{- \frac{3}{2}} \\ = \frac{1}{a^{\frac{3}{2}}} \\ = \frac{1}{\sqrt{a^{3}}} \end{matrix}$
$\begin{matrix} {(x^{\frac{2}{3}})}^{- 2} & = x^{- \frac{4}{3}} \\ = \frac{1}{x^{\frac{4}{3}}} \\ = \frac{1}{\sqrt[3]{x^{4}}} \\ = \frac{1}{\sqrt[3]{x^{3} . x}} \\ = \frac{1}{x \sqrt[3]{x}} \end{matrix}$
$\begin{matrix} {(\frac{a}{b})}^{- \frac{3}{2}} & = \frac{1}{{(\frac{a}{b})}^{\frac{3}{2}}} \\ = \frac{1}{\frac{a^{\frac{3}{2}}}{b^{\frac{3}{2}}}} \\ = \frac{b^{\frac{3}{2}}}{a^{\frac{3}{2}}} \\ = \frac{\sqrt{b^{3}}}{\sqrt{a^{3}}} \\ = \frac{\sqrt{b^{2} . b}}{\sqrt{a^{2} . a}} \\ = \frac{b \sqrt{b}}{a \sqrt{a}} \\ = \frac{b}{a} \sqrt{\frac{b}{a}} \end{matrix}$
$\begin{matrix} {(x^{- \frac{1}{2}})}^{\frac{1}{3}} & = x^{- \frac{1}{6}} \\ = \frac{1}{x^{\frac{1}{6}}} \\ = \frac{1}{\sqrt[6]{x}} \end{matrix}$
Expresar con exponentes positivos:
$\begin{matrix} \sqrt{a^{- 3}} & = a^{- \frac{3}{2}} \\ = \frac{1}{a^{\frac{3}{2}}} \end{matrix}$
$\begin{matrix} 2 \sqrt{x^{- 3} y^{- 4}} & = 2 x^{- \frac{3}{2}} y^{- \frac{}{2}} \\ = 2 x^{- \frac{3}{2}} y^{- 2} \\ = \frac{2}{x^{\frac{3}{2}} y^{2}} \end{matrix}$
$\begin{matrix} \frac{a^{\frac{2}{3}}}{\sqrt{x^{- 5}}} & = \frac{a^{\frac{2}{3}}}{x^{- \frac{5}{2}}} \\ = a^{\frac{2}{3}} x^{\frac{5}{2}} \end{matrix}$
$\begin{matrix} \frac{3 \sqrt[3]{m^{2}}}{5 \sqrt[4]{n^{- 3}}} & = \frac{3 m^{\frac{2}{3}}}{5 n^{- \frac{3}{4}}} \\ = \frac{3 m^{\frac{2}{3}} n^{\frac{3}{4}}}{5} \end{matrix}$
$\begin{matrix} a^{- \frac{3}{5}} \sqrt[4]{b^{- 3}} & = \frac{b^{- \frac{3}{4}}}{a^{\frac{3}{5}}} \\ = \frac{1}{a^{\frac{3}{5}} b^{\frac{3}{4}}} \end{matrix}$
$\begin{matrix} x^{2} \sqrt{x^{- 1}} & = x^{2} . x^{- \frac{1}{2}} \\ = x^{\frac{3}{2}} \end{matrix}$
$\begin{matrix} \frac{1}{\sqrt{a^{- 7} b^{- 6}}} & = \frac{1}{a^{- \frac{7}{2}} b^{- \frac{}{2}}} \\ = \frac{1}{a^{- \frac{7}{2}} b^{- 3}} \\ a^{\frac{7}{2}} b^{3} \end{matrix}$
$\begin{matrix} \frac{3 x^{- \frac{2}{3}}}{\sqrt{y^{- 4}}} & = \frac{3}{x^{\frac{2}{3}} y^{- \frac{}{2}}} \\ = \frac{3 y^{2}}{x^{\frac{2}{3}}} \end{matrix}$
$\begin{matrix} \sqrt{m^{- 1}} \sqrt[3]{n^{- 3}} & = m^{- \frac{1}{2}} n^{- \frac{3}{3}} \\ = \frac{1}{m^{\frac{1}{2}} n} \end{matrix}$
Hallar el valor de:
$\begin{matrix} 1 6^{\frac{3}{2}} & = \sqrt{1 6^{3}} \\ = \sqrt{1 6^{2} .16} \\ = 16 \sqrt{16} \\ = 16 (4) \\ = 64 \end{matrix}$
$\begin{matrix} 8^{\frac{2}{3}} & = \sqrt[3]{8^{2}} \\ = \sqrt[3]{{(2^{3})}^{2}} \\ = 2^{2} \\ = 4 \end{matrix}$
$\begin{matrix} 8 1^{\frac{3}{4}} & = \sqrt[4]{8 1^{3}} \\ = \sqrt[4]{{(3^{4})}^{3}} \\ = 3^{3} \\ = 27 \end{matrix}$
$\begin{matrix} 9^{- \frac{5}{2}} & = \frac{1}{9^{\frac{5}{2}}} \\ = \frac{1}{\sqrt{9^{5}}} \\ = \frac{1}{\sqrt{{(3^{2})}^{5}}} \\ = \frac{1}{3^{5}} \\ = \frac{1}{243} \end{matrix}$
$\begin{matrix} {(- 27)}^{\frac{2}{3}} & = \sqrt[3]{- 2 7^{2}} \\ = \sqrt[3]{{(- 3^{3})}^{2}} \\ = {(- 3)}^{2} \\ = 9 \end{matrix}$
$\begin{matrix} {(- 32)}^{\frac{2}{5}} & = \sqrt[5]{{(- 32)}^{2}} \\ = \sqrt[5]{{(- 2^{5})}^{2}} \\ = {(- 2)}^{2} \\ = 4 \end{matrix}$
$\begin{matrix} 4 9^{- \frac{3}{2}} & = \frac{1}{4 9^{\frac{3}{2}}} \\ = \frac{1}{\sqrt{4 9^{3}}} \\ = \frac{1}{\sqrt{{(7^{2})}^{3}}} \\ = \frac{1}{7^{3}} \\ = \frac{1}{343} \end{matrix}$
$\begin{matrix} {(\frac{4}{9})}^{\frac{5}{2}} & = \sqrt{{(\frac{4}{9})}^{5}} \\ = \sqrt{{(\frac{2^{2}}{3^{2}})}^{5}} \\ = \frac{2^{5}}{3^{5}} \\ = \frac{32}{243} \end{matrix}$
$\begin{matrix} {(\frac{8}{27})}^{- \frac{1}{3}} & = {(\frac{27}{8})}^{\frac{1}{3}} \\ = \sqrt[3]{\frac{27}{8}} \\ = \sqrt[3]{{(\frac{3}{2})}^{3}} \\ = \frac{3}{2} \end{matrix}$
$\begin{matrix} {(\frac{25}{36})}^{- \frac{1}{2}} & = {(\frac{36}{25})}^{\frac{1}{2}} \\ = \sqrt{\frac{36}{25}} \\ = \sqrt{{(\frac{6}{5})}^{2}} \\ = \frac{6}{5} \end{matrix}$
$\begin{matrix} {(\frac{32}{243})}^{- \frac{1}{5}} & = {(\frac{243}{32})}^{\frac{1}{5}} \\ = \sqrt[5]{\frac{243}{32}} \\ = \sqrt[5]{{(\frac{3}{2})}^{5}} \\ = \frac{3}{2} \end{matrix}$
$\begin{matrix} {(- \frac{27}{64})}^{- \frac{2}{3}} & = {(- \frac{64}{27})}^{\frac{2}{3}} \\ = \sqrt[3]{{(- \frac{64}{27})}^{2}} \\ = \sqrt[3]{{[{(- \frac{4}{3})}^{3}]}^{2}} \\ = {(- \frac{4}{3})}^{2} \\ = \frac{16}{9} \end{matrix}$
$\frac{1}{9^{- 3}} = 9^{3} = 729$
$\begin{matrix} {(\frac{16}{81})}^{- \frac{5}{4}} & = {(\frac{81}{16})}^{\frac{5}{4}} \\ = \sqrt[4]{{(\frac{81}{16})}^{5}} \\ = \sqrt[4]{{[{(\frac{3}{2})}^{4}]}^{5}} \\ = {(\frac{3}{2})}^{5} \\ = \frac{243}{32} \end{matrix}$
$\begin{matrix} {(- \frac{32}{243})}^{- \frac{2}{5}} & = {(- \frac{243}{32})}^{\frac{2}{5}} \\ = \sqrt[5]{{(- \frac{243}{32})}^{2}} \\ = \sqrt[5]{{[{(- \frac{3}{2})}^{5}]}^{2}} \\ = {(- \frac{3}{2})}^{2} \\ = \frac{9}{4} \end{matrix}$
$\begin{matrix} {(2 \frac{7}{9})}^{- \frac{3}{2}} & = {(\frac{25}{9})}^{- \frac{3}{2}} \\ = {(\frac{9}{25})}^{\frac{3}{2}} \\ = \sqrt{{(\frac{9}{25})}^{3}} \\ = \sqrt{{[{(\frac{3}{5})}^{2}]}^{3}} \\ = {(\frac{3}{5})}^{3} \\ = \frac{27}{125} \end{matrix}$
$\begin{matrix} {(5 \frac{1}{16})}^{- \frac{1}{4}} & = {(\frac{81}{16})}^{- \frac{1}{4}} \\ = {(\frac{16}{81})}^{\frac{1}{4}} \\ = \sqrt[4]{{(\frac{2}{3})}^{4}} \\ = \frac{2}{3} \end{matrix}$
$\begin{matrix} 8^{\frac{2}{3}} \times 4^{\frac{3}{2}} & = {(2^{3})}^{\frac{2}{3}} \times {(2^{2})}^{\frac{3}{2}} \\ = 2^{2} \times 2^{3} \\ = 2^{5} \\ = 32 \end{matrix}$
$\begin{matrix} 9^{\frac{5}{2}} \times 2 7^{- \frac{1}{3}} & = {(3^{2})}^{\frac{5}{2}} \times {(3^{3})}^{- \frac{1}{3}} \\ = 3^{5} \times 3^{- 1} \\ = 3^{4} \\ = 81 \end{matrix}$
$\begin{matrix} 24 3^{- \frac{1}{5}} \times 12 8^{\frac{3}{7}} & = {(3^{5})}^{- \frac{1}{5}} \times {(2^{7})}^{\frac{3}{7}} \\ = 3^{- 1} \times 2^{3} \\ = \frac{8}{3} \end{matrix}$