$a)\sqrt[3]{\sqrt{400}+7^1}=\sqrt[3]{20+7}=\sqrt[3]{27}=3$

$b)5^2:\frac{\sqrt{0,25}}{\sqrt[3]{0,008}}=25:\frac{0,5}{0,2}=25:\frac{5}{2}=25\cdot \frac{2}{5}=\frac{50}{5}=10$

$c)\frac{{\left[{\left(\frac{3}{4}\right)}^3\right]}^4\cdot {\left(4^4\right)}^3}{{\left(3^2\right)}^6}=\frac{{\left(\frac{3}{4}\right)}^{3\cdot 4}\cdot 4^{4\cdot 3}}{3^{2\cdot 6}}=\frac{{\left(\frac{3}{4}\right)}^{12}\cdot 4^{12}}{3^{12}}={\left(\frac{\frac{3}{4}\cdot 4}{3}\right)}^{12}={\left(\frac{3}{3}\right)}^{12}=1^{12}=1$