$\text{a)}\sqrt{a^4}=\sqrt{{\left(a^2\right)}^2}=a^2$  

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$\text{b)}\sqrt[5]{a^{30}}=\sqrt[5]{{\left(a^6\right)}^5}=a^6$  

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$\text{c)}\sqrt{4^n}=\sqrt{{\left(2^2\right)}^n}=\sqrt{{\left(2^n\right)}^2}=2^n$  

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$\text{d)}\sqrt[3]{a^{2}b}\cdot \sqrt[3]{a^4{b}^2}=\sqrt[3]{a^{2}b\cdot a^4{b}^2}=\sqrt[3]{a^6{b}^3}=\sqrt[3]{{\left(a^2\right)}^3\cdot b^3}=\sqrt[3]{{\left(a^{2}b\right)}^3}=a^{2}b$  

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$\text{e)}\frac{\sqrt{a^{2}b}}{\sqrt{b}}=\frac{\sqrt{a^2}\cdot \sqrt{b}}{\sqrt{b}}=\frac{\sqrt{a^2}\cdot \sqrt{b}}{\sqrt{b}}=\left|a\right|$  

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$\text{f)}{\left(\sqrt{b}\right)}^3={\left(\sqrt{b}\right)}^2\cdot \sqrt{b}=b\cdot \sqrt{b}=b\sqrt{b}$  

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$\text{g)}\left(\sqrt[4]{a^2}+\sqrt{b}\right)\left(-\sqrt[4]{a^2}-\sqrt{b}\right)=-\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)=-{\left(\sqrt{a}+\sqrt{b}\right)}^2$  

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$\text{h)}\sqrt[3]{3b^2}\cdot \sqrt[3]{9b^5}=\sqrt[3]{3b^2\cdot 9b^5}=\sqrt[3]{3^3\cdot b^7}=\sqrt[3]{3^3}\cdot \sqrt[3]{b^6\cdot b}=3\cdot \sqrt[3]{{\left(b^2\right)}^3\cdot b}=$  

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

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$\text{i)}\sqrt{a^2+a^2+a^2+a^2}=\sqrt{4\cdot a^2}=\sqrt{4}\cdot \sqrt{a^2}=2\cdot \left|a\right|$