$a){\left(\sqrt{10}\right)}^6={\left({10}^{\frac{1}{2}}\right)}^6={10}^{\frac{1}{2}\cdot 6}={10}^3=1000$

$b)\sqrt{4^{10}}={\left(4^{10}\right)}^{\frac{1}{2}}=4^{10\cdot \frac{1}{2}}=4^5=1024$

$c)\sqrt[3]{{125}^2}={\sqrt[3]{125}}^2=5^2=25$

$d)\sqrt[5]{{32}^3}={\sqrt[5]{32}}^3=2^3=8$

$e){\left(\sqrt[8]{625}\right)}^2={\left({625}^{\frac{1}{8}}\right)}^2={625}^{\frac{2}{8}}={625}^{\frac{1}{4}}={\left(5^4\right)}^{\frac{1}{4}}=5^{4\cdot \frac{1}{4}}=5$

$f)\sqrt{\sqrt{81}}=\sqrt{9}=3$  

$g){\left(\sqrt[3]{\sqrt{27}}\right)}^2={\left(\sqrt[3]{{27}^{\frac{1}{2}}}\right)}^2={\left({\left({27}^{\frac{1}{2}}\right)}^{\frac{1}{3}}\right)}^2={\left({27}^{\frac{1}{6}}\right)}^2={27}^{\frac{2}{6}}={27}^{\frac{1}{3}}={\left(3^3\right)}^{\frac{1}{3}}=3^1=3$  

$h)\sqrt[3]{\sqrt{4^3}}=\sqrt[3]{\sqrt{64}}=\sqrt[3]{8}=2$  

$i){\left(\sqrt{\sqrt[3]{49}}\right)}^3={\left(\sqrt{{49}^{\frac{1}{3}}}\right)}^3={\left({\left({49}^{\frac{1}{3}}\right)}^{\frac{1}{2}}\right)}^3={\left({49}^{\frac{1}{6}}\right)}^3={49}^{\frac{3}{6}}={49}^{\frac{1}{2}}=\sqrt{49}=7$  

$j)\sqrt[3]{\sqrt[4]{{16}^6}}=\sqrt[3]{{16}^{\frac{6}{4}}}=\sqrt[3]{{16}^{\frac{3}{2}}}={\left({16}^{\frac{3}{2}}\right)}^{\frac{1}{3}}={16}^{\frac{3}{6}}={16}^{\frac{1}{2}}=\sqrt{16}=4$