Question

De preferencia a 2a.

Answer 1

$a)~~\boxed{\frac{b^4}{a^6}};~~b)~~\boxed{\frac{1}{15.625}}~~c)~~\boxed{\frac{2.048}{27}}$

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Para responder esse exercício é necessário saber potenciação e suas propriedades e operações com frações.

Propriedades que iremos utilizar:

• Divisão de potência de mesma base;
• Multiplicação de potências de mesmo expoente e bases diferentes;
• Potência de potência;
• Potência com expoente negativo;
• Divisão de frações;
• Multiplicação de frações.

2) Simplifique as expressões abaixo:

$a)~~\Large\displaystyle\text{ \begin{gathered}\tt~\left(\frac{ab}{a^3}\right)^2\div\left(\frac{a^{2}b}{b^3}\right)\end{gathered} }\\\\\\\large\displaystyle\text{ \begin{gathered}\tt~\left(\frac{\backslash\!\!\!ab}{a^{\backslash\!\!\!3}}\right)^2~\Rightarrow~\frac{a^1}{a^3}~\Rightarrow~a^{3-1}~\Rightarrow~\boxed{a^{2}}~\checkmark\end{gathered} }\\\\\\\large\displaystyle\text{ \begin{gathered}\tt~\left(\frac{a^2\diagup\!\!\!\!b}{b^{\diagup\!\!\!\!3}}\right)~\Rightarrow~\frac{b^1}{b^3}~\Rightarrow~b^{3-1}~\Rightarrow~\boxed{b^2}~\checkmark\end{gathered} }\\\\\\\large\displaystyle\begin{gathered}\tt~\left(\frac{b}{a^2}\right)^2\div\left(\frac{a^2}{b^2}\right)\end{gathered}\\\\\\\large\displaystyle\text{ \begin{gathered}\tt~\left(\frac{b}{a^2}\right)^2~\Rightarrow~\frac{b^2}{(a^2)^2}~\Rightarrow~\frac{b^2}{a^{2\cdot2}}~\Rightarrow~\boxed{\frac{b^2}{a^4}}~\checkmark\end{gathered} }\\\\\\\large\displaystyle\text{ \begin{gathered}\tt~\frac{b^2}{a^4}\div\frac{a^2}{b^2}~\Rightarrow~\frac{b^2}{a^4}\cdot\frac{b^2}{a^2}~\Rightarrow~\frac{b^2\cdot\,\!b^2}{a^4\cdot\,\!b^2}~\Rightarrow~\frac{a^{2+2}}{a^{4+2}}~\Rightarrow~\boxed{\frac{b^4}{a^6}}~\checkmark~\hookleftarrow~\underline{Resposta}\end{gathered} }$

$b)~~\large\displaystyle\text{ \begin{gathered}\frac{5\cdot25^2\cdot125^{-2}}{3.125}\end{gathered} }\\\\\\\large\displaystyle\text{ \begin{gathered}\frac{5\cdot5^4\cdot5^{-6}}{5^5}~\Rightarrow~5^1\cdot5^4\cdot5^{-6}~\Rightarrow~5^{1+4-6}~\Rightarrow~\boxed{5^{-1}}~\checkmark\end{gathered} }\\\\\\\large\displaystyle\text{ \begin{gathered}\frac{5^{-1}}{5^5}~\Rightarrow~\frac{1}{5^{5-(-1)}}~\Rightarrow~\frac{1}{5^{5+1}}~\Rightarrow~\boxed{\frac{1}{5^6}}~\checkmark\end{gathered} }\\\\\\\large\displaystyle\text{ \begin{gathered}\frac{1}{5^6}~\Rightarrow~\boxed{\frac{1}{15.625}}~\checkmark~\hookleftarrow~\underline{Resposta}\end{gathered} }$

$c)~~ \large\displaystyle\text{ \begin{gathered}\sf\frac{2^2\cdot(3\cdot2)^2\cdot18^{-2}}{3\cdot8^{-3}}\end{gathered} }\\\\\\\frac{2^2\cdot9\cdot4\cdot18^{-2}}{3\cdot8^{-3}}~\Rightarrow~8^{-3}~\Rightarrow~\left(2^3\right)^{-3}~\Rightarrow~\left(2^3\right)^{3\cdot(-3)}~\Rightarrow~\boxed{2^{-9}}~\checkmark\\\\\\\frac{2^2\diagup\!\!\!\!2\cdot9\cdot4\cdot18^{-2}}{3\cdot\diagup\!\!\!\!2^{-\diagup\!\!\!\!9}}~\Rightarrow~\frac{2^2}{2^{-9}}~\Rightarrow~2^{2-(-9)}~\Rightarrow~2^{2+9}~\Rightarrow~\boxed{2^{11}}~\checkmark\\\\\\\frac{2^{11}\cdot\diagup\!\!\!\!9\cdot4\cdot18{-2}}{\diagup\!\!\!\!3}~\Rightarrow~2^{11}\cdot3\cdot4\cdot18^{-2}\\\\\\18^{-2}~\Rightarrow~\boxed{a^{-n}=\frac{1}{a^{n}}}\bigstar~\Rightarrow~\boxed{\frac{1}{18^2}}\\\\\\2^{11}\cdot3\cdot2^2\cdot\frac{1}{18^2}\\\\\\18^2~\Rightarrow~(9\cdot2)^2~\Rightarrow~9^2\cdot2^2~\Rightarrow~\boxed{3^4\cdot2^2}~\checkmark\\\\\\2^{11}\cdot3\cdot\diagup\!\!\!\!2^{\diagup\!\!\!\!2}\cdot\frac{1}{3^4\cdot\diagup\!\!\!\!2^{\diagup\!\!\!\!2}}~\Rightarrow~2^{11}\cdot\diagup\!\!\!3\cdot\frac{1}{3^{\diagup\!\!\!\!4}}~\Rightarrow~\frac{3^1}{1}\cdot\frac{1}{3^4}~\Rightarrow~3^{4-1}~\Rightarrow~\boxed{3^3}~\checkmark\\\\\\2^{11}\cdot\frac{1}{3^3}~\Rightarrow~2^{11}\cdot\frac{1}{27}~\Rightarrow~\frac{2^{11}}{27}~\Rightarrow~\boxed{\frac{2.048}{27}}~\checkmark\hookleftarrow\underline{Resposta}$

$\mathscr{By:~\blue{DANIEL}}$

Answer 2

Resposta:

b) 5 . 25².125- ²

Explicação passo-a-passo:

espero ter ajudado vc