$a)$    $4^{\frac{1}{2}}=\sqrt{4}=2$  

$4^{\frac{3}{2}}=\sqrt{4^3}={\left(\sqrt{4}\right)}^3=2^3=8$  

$4^{\frac{5}{2}}=\sqrt{4^5}={\left(\sqrt{4}\right)}^5=2^5=32$  

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$b)$    $8^{\frac{1}{3}}=\sqrt[3]{8}=2$  

$8^{\frac{2}{3}}=\sqrt[3]{8^2}={\left(\sqrt[3]{8}\right)}^2=2^2=4$  

$8^{\frac{4}{3}}=\sqrt[3]{8^4}={\left(\sqrt[3]{8}\right)}^4=2^4=16$  

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

${27}^{\frac{2}{3}}=\sqrt[3]{{27}^2}=\sqrt[3]{{\left(3^3\right)}^2}=\left(\sqrt[3]{3^6}\right)={\left(\sqrt[3]{3}\right)}^6={\left(\sqrt[3]{3}\right)}^{3\cdot 2}={\left({\left(\sqrt[3]{3}\right)}^3\right)}^2)=3^2=9$  

${27}^{1\frac{1}{3}}={27}^{\frac{4}{3}}=\sqrt[3]{{27}^4}=\sqrt[3]{{\left(3^3\right)}^4}=\left(\sqrt[3]{3^{12}}\right)={\left(\sqrt[3]{3}\right)}^{12}={\left(\sqrt[3]{3}\right)}^{3\cdot 4}={\left({\left(\sqrt[3]{3}\right)}^3\right)}^4=3^4=81$  

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$d)$    ${81}^{\frac{1}{4}}=\sqrt[4]{81}=\sqrt[4]{9^2}=\sqrt[4]{{\left(3^2\right)}^2}=\sqrt[4]{3^4}={\left(\sqrt[4]{3}\right)}^4=3$  

${81}^{\frac{3}{4}}=\sqrt[4]{{81}^3}=\sqrt[4]{{\left(9^2\right)}^3}=\sqrt[4]{9^6}=\sqrt[4]{{\left(3^2\right)}^6}=\sqrt[4]{3^{12}}={\left(\sqrt[4]{3}\right)}^{12}={\left(\sqrt[4]{3}\right)}^{4\cdot 3}={\left({\left(\sqrt[4]{3}\right)}^4\right)}^3)=3^3=27$  

${81}^{-\frac{1}{2}}=\sqrt{{81}^{-1}}=\sqrt{{\left(9^2\right)}^{-1}}=\sqrt{9^{-2}}=\sqrt{{\left(3^2\right)}^{-2}}=\sqrt{3^{-4}}={\left(\sqrt{3}\right)}^{-4}={\left(\sqrt{3}\right)}^{2\cdot \left(-2\right)}={\left({\left(\sqrt{3}\right)}^2\right)}^{-2}=3^{-2}={\left(\frac{1}{3}\right)}^2=\frac{1}{9}$  

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$e)$    ${\left(\frac{1}{16}\right)}^{-\frac{1}{4}}={16}^{\frac{1}{4}}=\sqrt[4]{16}=\sqrt[4]{2^4}={\sqrt[4]{2}}^4=2$  

${\left(\frac{25}{81}\right)}^{-\frac{1}{2}}={\left(\frac{81}{25}\right)}^{\frac{1}{2}}=\sqrt{\frac{81}{25}}=\frac{\sqrt{81}}{\sqrt{25}}=\frac{9}{5}=1\frac{4}{5}$  

${\left(\frac{8}{27}\right)}^{-1\frac{1}{3}}={\left(\frac{8}{27}\right)}^{-\frac{4}{3}}={\left(\frac{27}{8}\right)}^{\frac{4}{3}}=\sqrt[3]{{\left(\frac{27}{8}\right)}^4}={\left(\sqrt[3]{\left(\frac{27}{8}\right)}\right)}^4={\left(\frac{\sqrt[3]{27}}{\sqrt[3]{8}}\right)}^4=\frac{{\left(\sqrt[3]{27}\right)}^4}{{\left(\sqrt[3]{8}\right)}^4}=\frac{3^4}{2^4}=\frac{81}{16}=5\frac{1}{16}$  

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$f)$    ${\left(2\frac{1}{4}\right)}^{\frac{1}{2}}={\left(\frac{9}{4}\right)}^{\frac{1}{2}}=\sqrt{\frac{9}{4}}=\frac{\sqrt{9}}{\sqrt{4}}=\frac{3}{2}=1\frac{1}{2}$  

${\left(3\frac{3}{8}\right)}^{-\frac{1}{3}}={\left(\frac{27}{8}\right)}^{-\frac{1}{3}}={\left(\frac{8}{27}\right)}^{\frac{1}{3}}=\sqrt[3]{\frac{8}{27}}=\frac{\sqrt[3]{8}}{\sqrt[3]{27}}=\frac{2}{3}$  

${\left(1\frac{7}{9}\right)}^{\frac{3}{2}}={\left(\frac{16}{9}\right)}^{\frac{3}{2}}=\sqrt{{\left(\frac{16}{9}\right)}^3}={\left(\sqrt{\frac{16}{9}}\right)}^3={\left(\frac{4}{3}\right)}^3=\frac{4^3}{3^3}=\frac{64}{27}=2\frac{10}{27}$