$\text{a)}{\left(\sqrt{7}\right)}^3={\left(\sqrt{7}\right)}^{2+1}={\left(\sqrt{7}\right)}^2\cdot \sqrt{7}=7\sqrt{7}$  

$\text{b)}{\left(\sqrt[3]{5}\right)}^4={\left(\sqrt[3]{5}\right)}^{3+1}={\left(\sqrt[3]{5}\right)}^3\cdot \sqrt[3]{5}=5\sqrt[3]{5}$  

$\text{c)}{\left(\sqrt{3}\right)}^5={\left(\sqrt{3}\right)}^{2+2+1}={\left(\sqrt{3}\right)}^2\cdot {\left(\sqrt{3}\right)}^2\cdot \sqrt{3}=3\cdot 3\cdot \sqrt{3}=9\sqrt{3}$  

$\text{d)}{\left(\sqrt[3]{2}\right)}^8={\left(\sqrt[3]{2}\right)}^{3+3+2}={\left(\sqrt[3]{2}\right)}^3\cdot {\left(\sqrt[3]{2}\right)}^3\cdot {\left(\sqrt[3]{2}\right)}^2=2\cdot 2\cdot {\left(\sqrt[3]{2}\right)}^2=4{\left(\sqrt[3]{2}\right)}^2=$       

$=4\cdot \sqrt[3]{2}\cdot \sqrt[3]{2}=4\cdot \sqrt[3]{2\cdot 2}=4\sqrt[3]{4}$  

$\text{e)}{\left(\sqrt{\sqrt{5}}\right)}^6={\left({\left(\sqrt{\sqrt{5}}\right)}^2\right)}^3={\left(\sqrt{5}\right)}^3=5\sqrt{5}$  

$\text{f)}{\left(\sqrt[3]{\sqrt[3]{10}}\right)}^6={\left({\left(\sqrt[3]{\sqrt[3]{10}}\right)}^3\right)}^2={\left(\sqrt[3]{10}\right)}^2=\sqrt[3]{10}\cdot \sqrt[3]{10}=\sqrt[3]{10\cdot 10}=\sqrt[3]{100}$  

$\text{g)}{\left(\sqrt{\sqrt[3]{6}}\right)}^4={\left({\left(\sqrt{\sqrt[3]{6}}\right)}^2\right)}^2={\left(\sqrt[3]{6}\right)}^2=\sqrt[3]{6}\cdot \sqrt[3]{6}=\sqrt[3]{6\cdot 6}=\sqrt[3]{36}$  

$\text{h)}{\left(\sqrt[3]{\sqrt{2}}\right)}^9={\left({\left(\sqrt[3]{\sqrt{2}}\right)}^3\right)}^3={\left(\sqrt{2}\right)}^3=2\sqrt{2}$