Question

Calcule e dê o resultado na forma mais simples.
( (8^5/6 . 8^3/2 ) / (8^10/6 ) )^-1 : ³√ 0,008

Answer 1

$\frac{(8^{\frac{5}{6}} * 8^{\frac{3}{2}})^{-1}}{ \sqrt[3]{0,008} } \\\\ Aplicando\ a\ propriedade\ da\ potencia\c{c}\~ao\ bases\ iguais\ expoentes\\ diferentes: (a^m*a^n=a^{m+n}) \\\\\frac{(8^{\frac{5}{6}+\frac{3}{2}})^{-1}}{ \sqrt[3]{0,008} } \\\\\\\ Tirando\ o\ MMC \to \left\{\begin{array}{ccc}2,6|2 \\ 1,3|3 \\ 1,1|\ \ \\\\ MMC \to 2^1*3^1=2*3=6\end{array}\right \\\\\\\frac{(8^{\frac{5}{6}+\frac{(3*3)}{6}})^{-1}}{ \sqrt[3]{0,008} }} \\\\ \frac{(8^{\frac{5}{6}+\frac{9}{6}})^{-1}}{ \sqrt[3]{0,008} }}$
$\\\\ \frac{(8^{\frac{14}{6}})^{-1}}{ \sqrt[3]{0,008} } = \frac{(8^{\frac{14}{6}})^{-1}}{1}*\frac{1}{ {\sqrt[3]{0,008}}} \\\\ \frac{(8^{\frac{14}{6}})^{-1}}{1}*\frac{1}{ {\sqrt[3]{0,008}}} \\\\ \frac{1}{8^{\frac{14}{6}}}*\frac{1}{ {\sqrt[3]{0,008}}} = \frac{1}{8^{\frac{14}{6}} * \sqrt[3]{0,008}} \\\\\\ \begin{array}{ccc}\boxed{\boxed{0,008 = 0,2^3}}\end{array}\right \\\\\\ \frac{1}{8^{\frac{14}{6}} * \sqrt[3]{0,008}} = \frac{1}{8^{\frac{14}{6}} * \sqrt[3]{0,2^3}}$
$\frac{1}{8^{\frac{14}{6}} * \sqrt[3]{0,2^3}} = \frac{1}{8^{\frac{14}{6}} * 0,2} = \frac{1}{8^{\frac{14}{6}}} * \frac{1}{0,2} = \frac{1}{8^{\frac{14}{6}}} * \frac{1}{ \frac{1}{5} } \\\\\\ \left\{\begin{array}{ccc}Divis\~ao\ de\ fra\c{c}\~oes \\\\ \frac{1}{1} : \frac{1}{5} = \frac{1}{1} * \frac{5}{1} = 5 \\\\ Portanto, \\\\\boxed{\boxed{ \frac{1}{ \frac{1}{5} } = 5}}\end{array}\right \\\\\\ \frac{1}{8^{\frac{14}{6}}} * \frac{1}{ \frac{1}{5} } = \frac{1}{8^{\frac{14}{6}}} * \frac{5}{1} }$
$\frac{1}{8^{\frac{14}{6}}} * \frac{5}{1} } = \frac{5}{8^{\frac{14}{6}}} \\\\\\ \left\{\begin{array}{ccc}8^{\frac{14}{6}} = \sqrt[6]{8^{14}} \\\\ \sqrt[6]{8^{14}} = \sqrt[6]{2^{3(14)}} \\\\ \sqrt[6]{2^{3(14)}} = \sqrt[6]{2^{42}} \\\\ \sqrt[6]{2^{42}} = 2^{ \frac{42}{6} \\\\2^{ \frac{42}{6}} = 2^7 \\\\Portanto,\\\\\boxed{\boxed{8^{\frac{14}{6}} = 2^7}}\end{array}\right \\\\\\ \frac{5}{8^{\frac{14}{6}}} = \frac{5}{2^{7}} = \frac{5}{128} \\\\ N\~ao\ da\ para\ simplificar\ mais\ do\ que\ isso,\ ent\~ao...$

$Resposta: \boxed{\boxed{ \frac{5}{2^7} }}$

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Answer 2

( (8^5/6 . 8^3/2 ) /(8^10/6 ) )^-1 :  ³√ 0,008

    ( 8^14/6 / 8^10/6 ) ^-1 ==> (8^(14/6-10/6)^-1
             0,2                               0,2

 (8^(4/6)^-1 ==> (2^3^(4/6)^-1
         0,2                           0,2                        

(2^12/6)^-1  ==> (2^2)^-1  . 10   ==>    5    ==> 5
      2/10                                2             2^2         4

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separei cada questão para melhor entender 


( (8^5/6 . 8^3/2 ) = 8^(5/6+3/2) ==> 8^( 5+9)/6 ==> 8^14/6 ==> 8^14/6 
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 (8^10/6 ) 
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 ³√ 0,008 ==>∛0,2³ ==> 0,2