Question

Me pueden dar y explicar la respuesta? ​

Answer 1

Respuesta:

$\left[8(4/5)^{-2}-(2/3)^{-3}-(8/9)^{-1}\right]^{\left(\frac{3}{\sqrt[3]{9}}\right)^{-3}}=2$

Explicación paso a paso:

$\left[8(4/5)^{-2}-(2/3)^{-3}-(8/9)^{-1}\right]^{\left(\frac{3}{\sqrt[3]{9}}\right)^{-3}}\,\,\texttt{pero:}\,\,a^{-n}=\frac{1}{a^{n}}\\\left(\sqrt[n]{a}\right)^{n}=a^{\frac{1}{n}*n}=a\,\,\texttt{lo cual implica que la expresi\'on inicial, equivale a:}\\\\\left[\frac{8}{\left(\frac{4}{3}\right)^{2}}-\frac{1}{\left(\frac{2}{3}\right)^{3}}-\frac{1}{\left(\frac{8}{9}\right)}\right]^{\frac{1}{\left(\frac{3}{\sqrt[3]{9}}\right)^{3}}}\,\,\texttt{recordemos tambi\'en que:}\,\,\frac{1}{\left(\frac{a}{b}\right)^{m}}=\frac{1}{\frac{a^{m}}{b^{m}}}\,\,\texttt{en consecuencia:}\\\\\frac{8}{\left(\frac{4}{5}\right)^{2}}=\frac{8}{\frac{4^{2}}{5^{2}}}=\frac{8}{\frac{16}{25}}=\frac{(8)(25)}{(16)(8)}=\frac{25}{2}\\\\\frac{1}{\left(\frac{2}{3}\right)^{3}}=\frac{1}{\frac{2^{3}}{3^{3}}}=\frac{1}{\frac{8}{27}}=\frac{27}{8}\\\\\frac{1}{\frac{8}{9}}=\frac{9}{8}\\\\\frac{1}{\left(\frac{3}{\sqrt[3]{9}}\right)^{3}}=\frac{1}{\frac{3^{3}}{(\sqrt[3]{9})^{3}}}=\frac{1}{\frac{27}{9}}=\frac{9}{27}=\frac{1}{3}$

$\texttt{reuniendo todo nos queda:}$

$ex]\left[8(4/5)^{-2}-(2/3)^{-3}-(8/9)^{-1}\right]^{\left(\frac{3}{\sqrt[3]{9}}\right)^{-3}}=\left[\frac{25}{2}-\frac{27}{8}-\frac{9}{8}\right]^{\frac{1}{3}}=\\\\=\left[\frac{25}{2}-\frac{9}{2}\right]^{\frac{1}{3}}=\left[8\right]^{\frac{1}{3}}=\sqrt[3]{8}=2$