$\text{a)}{\left(7+{\left(2+4^{0,5}\right)}^{0,5}\right)}^{0,5}={\left(7+{\left(2+\sqrt{4}\right)}^{0,5}\right)}^{0,5}={\left(7+{\left(2+2\right)}^{0,5}\right)}^{0,5}={\left(7+4^{0,5}\right)}^{0,5}={\left(7+\sqrt{4}\right)}^{0,5}={\left(7+2\right)}^{0,5}=9^{0,5}=\sqrt{9}=3$  

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$\text{b)}{\left(5-{\left(4-9^{0,5}\right)}^{0,5}\right)}^{0,5}={\left(5-{\left(4-\sqrt{9}\right)}^{0,5}\right)}^{0,5}={\left(5-{\left(4-3\right)}^{0,5}\right)}^{0,5}={\left(5-1^{0,5}\right)}^{0,5}={\left(5-\sqrt{1}\right)}^{0,5}={\left(5-1\right)}^{0,5}=4^{0,5}=\sqrt{4}=2$  

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$\text{c)}{\left(\sqrt[7]{9\sqrt[6]{9}}\right)}^3={\left(\sqrt[7]{9\cdot 9^{\frac{1}{6}}}\right)}^3={\left(\sqrt[7]{9^{1\frac{1}{6}}}\right)}^3={\left(\sqrt[7]{9^{\frac{7}{6}}}\right)}^3={\left(9^{\frac{1}{6}}\right)}^3=9^{\frac{1}{2}}=\sqrt{9}=3$  

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$\text{d)}{\left(\sqrt[6]{8\sqrt[5]{8}}\right)}^{\frac{5}{3}}={\left(\sqrt[6]{8\cdot 8^{\frac{1}{5}}}\right)}^{\frac{5}{3}}={\left(\sqrt[6]{8^{1\frac{1}{5}}}\right)}^{\frac{5}{3}}={\left(\sqrt[6]{8^{\frac{6}{5}}}\right)}^{\frac{5}{3}}={\left(8^{\frac{1}{5}}\right)}^{\frac{5}{3}}=8^{\frac{1}{3}}=\sqrt[3]{8}=2$