$\text{a)}{\left(\frac{1}{4}\right)}^x=32$  

${\left({\left(2\right)}^{-2}\right)}^x=2^5$  

$2^{-2x}=2^5$  

$-2x=5$  

$x=-\frac{5}{2}$  

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$\text{b)}{\left(\frac{2}{\sqrt{2}}\right)}^x=\sqrt[3]{4}$  

${\left(2\cdot 2^{-\frac{1}{2}}\right)}^x=2^{\frac{2}{3}}$  

${\left(2^{\frac{1}{2}}\right)}^x=2^{\frac{2}{3}}$  

$2^{\frac{1}{2}x}=2^{\frac{2}{3}}$  

$\frac{1}{2}x=\frac{2}{3}$  

$x=\frac{4}{3}$  

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$\text{c)}{\left(\frac{16}{10}\right)}^x=\frac{25}{64}$  

${\left(\frac{8}{5}\right)}^x={\left(\frac{8}{5}\right)}^{-2}$  

$x=-2$  

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$\text{d)}{\left(\frac{2}{5}\right)}^x=\frac{5}{2}\cdot \frac{25}{4}$  

${\left(\frac{2}{5}\right)}^x=\frac{125}{8}$  

${\left(\frac{2}{5}\right)}^x={\left(\frac{2}{5}\right)}^{-3}$  

$x=-3$