$\text{a)}$  

$\sqrt{\sqrt{81}}=\sqrt{\sqrt{9^2}}=\sqrt{9}=\sqrt{3^2}=3$  

$\sqrt{\sqrt{0,0001}}=\sqrt{\sqrt{{\left(0,01\right)}^2}}=\sqrt{0,01}=\sqrt{{\left(0,1\right)}^2}=0,1$  

$\sqrt{\sqrt{\frac{16}{81}}}=\sqrt{\sqrt{\left(\frac{4^2}{9^2}\right)}}=\sqrt{\sqrt{{\left(\frac{4}{9}\right)}^2}}=\sqrt{\frac{4}{9}}=\sqrt{\frac{2^2}{3^2}}=\sqrt{{\left(\frac{2}{3}\right)}^2}=\frac{2}{3}$  

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$\text{b)}$  

$\sqrt{5\sqrt{25}}=\sqrt{5\cdot \sqrt{5^2}}=\sqrt{5\cdot 5}=\sqrt{5^2}=5$  

$\sqrt{12\sqrt{\frac{1}{9}}}=\sqrt{12\cdot \sqrt{\frac{1}{3^2}}}=\sqrt{12\cdot \sqrt{{\left(\frac{1}{3}\right)}^2}}=\sqrt{12\cdot \frac{1}{3}}=\sqrt{4}=\sqrt{2^2}=2$  

$\sqrt{\sqrt{2\sqrt{64}}}=\sqrt{\sqrt{2\cdot \sqrt{8^2}}}=\sqrt{\sqrt{2\cdot 8}}=\sqrt{\sqrt{16}}=\sqrt{\sqrt{4^2}}=\sqrt{4}==\sqrt{2^2}=2$  

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$\text{c)}$  

$\sqrt{\sqrt[3]{\frac{1}{64}}}=\sqrt{\sqrt[3]{\frac{1}{4^3}}}=\sqrt{\sqrt[3]{{\left(\frac{1}{4}\right)}^3}}=\sqrt{\frac{1}{4}}=\sqrt{\frac{1}{2^2}}=\sqrt{{\left(\frac{1}{2}\right)}^2}=\frac{1}{2}$  

$\sqrt[3]{\sqrt{1000000}}=\sqrt[3]{\sqrt{{1000}^2}}=\sqrt[3]{1000}=\sqrt[3]{{10}^3}=10$  

$\sqrt{\sqrt[3]{7^6}}=\sqrt{\sqrt[3]{{\left(7^2\right)}^3}}=\sqrt{7^2}=7$  

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$\text{d)}$  

$\sqrt[3]{3\sqrt{81}}=\sqrt[3]{3\cdot \sqrt{9^2}}=\sqrt[3]{3\cdot 9}=\sqrt[3]{\cdot }\left(27\right)=\sqrt[3]{3^3}=3$  

$\sqrt{10\sqrt[3]{1000}}=\sqrt{10\cdot \sqrt[3]{{10}^3}}=\sqrt{10\cdot 10}=\sqrt{{10}^2}=10$  

$\sqrt[3]{20\sqrt{\frac{4}{25}}}=\sqrt[3]{20\cdot \sqrt{\frac{2^2}{5^2}}}=\sqrt[3]{20\cdot \sqrt{{\left(\frac{2}{5}\right)}^2}}=\sqrt[3]{20\cdot \frac{2}{5}}=\sqrt[3]{8}=\sqrt[3]{2^3}=2$