$\text{a)}x={\left(6\frac{1}{4}\right)}^{0,5}\cdot {100}^{-0,5}={\left(\frac{25}{4}\right)}^{0,5}\cdot {\left(\frac{1}{100}\right)}^{0,5}=\sqrt{\frac{25}{4}}\cdot \sqrt{\frac{1}{100}}=\frac{5}{2}\cdot \frac{1}{10}=\frac{1}{4}$  

Liczba odwrotna do x to

$\frac{1}{x}=\frac{1}{\frac{1}{4}}=4\in \mathbf{N}$  

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$\text{b)}x={\left(5\frac{1}{16}\right)}^{\frac{1}{4}}\cdot 9^{-\frac{1}{2}}={\left(\frac{81}{16}\right)}^{\frac{1}{4}}\cdot {\left(\frac{1}{9}\right)}^{\frac{1}{2}}=\sqrt[4]{\frac{81}{16}}\cdot \sqrt{\frac{1}{9}}=\frac{3}{2}\cdot \frac{1}{3}=\frac{1}{2}$  

Liczba odwrotna do x to

$\frac{1}{x}=\frac{1}{\frac{1}{2}}=2\in \mathbf{N}$  

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$\text{c)}x={32}^{-0,5}\cdot {0,5}^{0,5}={\left(\frac{1}{32}\right)}^{0,5}\cdot {\left(\frac{1}{2}\right)}^{0,5}={\left(\frac{1}{32}\cdot \frac{1}{2}\right)}^{0,5}={\left(\frac{1}{64}\right)}^{0,5}=\sqrt{\frac{1}{64}}=\frac{1}{8}$  

Liczba odwrotna do x to

$\frac{1}{x}=\frac{1}{\frac{1}{8}}=8\in \mathbf{N}$  

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$\text{d)}x={54}^{-\frac{1}{3}}\cdot {0,25}^{\frac{1}{3}}={\left(\frac{1}{54}\right)}^{\frac{1}{3}}\cdot {\left(\frac{1}{4}\right)}^{\frac{1}{3}}={\left(\frac{1}{54}\cdot \frac{1}{4}\right)}^{\frac{1}{3}}={\left(\frac{1}{216}\right)}^{\frac{1}{3}}=\sqrt[3]{\frac{1}{216}}=\frac{1}{6}$  

Liczba odwrotna do x to

$\frac{1}{x}=\frac{1}{\frac{1}{6}}=6\in \mathbf{N}$