a)

$\frac{a^{\frac{7}{8}}}{\sqrt[4]{a^3}}=\frac{a^{\frac{7}{8}}}{a^{\frac{3}{4}}}=a^{\frac{7}{8}-\frac{3}{4}}=a^{\frac{7}{8}-\frac{6}{8}}=a^{\frac{1}{8}}$

$\frac{a^{\frac{5}{6}}}{\sqrt[3]{a^2}}=\frac{a^{\frac{5}{6}}}{a^{\frac{2}{3}}}=a^{\frac{5}{6}-\frac{2}{3}}=a^{\frac{5}{6}-\frac{4}{6}}=a^{\frac{1}{6}}$

$\frac{\sqrt[3]{x^2}}{x^{\frac{1}{2}}}=\frac{x^{\frac{2}{3}}}{x^{\frac{1}{2}}}=x^{\frac{2}{3}-\frac{1}{2}}=x^{\frac{4}{6}-\frac{3}{6}}=x^{\frac{1}{6}}$

$\frac{\sqrt[4]{x^3}}{x^{\frac{1}{3}}}=\frac{x^{\frac{3}{4}}}{x^{\frac{1}{3}}}=x^{\frac{3}{4}-\frac{1}{3}}=x^{\frac{9}{12}-\frac{4}{12}}=x^{\frac{5}{12}}$

$\frac{\sqrt[5]{x^2}}{x^{\frac{7}{20}}}=\frac{x^{\frac{2}{5}}}{x^{\frac{7}{20}}}=x^{\frac{2}{5}-\frac{7}{20}}=x^{\frac{8}{20}-\frac{7}{20}}=x^{\frac{1}{20}}$

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 b)

$a^{\frac{1}{2}}{b}^{\frac{1}{2}}={\left(ab\right)}^{\frac{1}{2}}$

$a^{\frac{3}{4}}{b}^{\frac{3}{4}}={\left(ab\right)}^{\frac{3}{4}}$   

$8^{\frac{4}{5}}\cdot 4^{\frac{4}{5}}={\left(8\cdot 4\right)}^{\frac{4}{5}}={32}^{\frac{4}{5}}=2^4=16$

${27}^{\frac{3}{4}}\cdot 3^{\frac{3}{4}}={\left(27\cdot 3\right)}^{\frac{3}{4}}={81}^{\frac{3}{4}}=3^3=27$   

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c)

$\frac{a^{\frac{7}{8}}}{b^{\frac{7}{8}}}={\left(\frac{a}{b}\right)}^{\frac{7}{8}}$

$\frac{x^{-\frac{5}{6}}}{y^{-\frac{5}{6}}}={\left(\frac{x}{y}\right)}^{-\frac{5}{6}}={\left(\frac{y}{x}\right)}^{\frac{5}{6}}$

$\frac{{64}^{\frac{2}{3}}}{8^{\frac{2}{3}}}={\left(\frac{64}{8}\right)}^{\frac{2}{3}}=8^{\frac{2}{3}}=2^2=4$

$\frac{{128}^{\frac{4}{5}}}{4^{\frac{4}{5}}}={\left(\frac{128}{4}\right)}^{\frac{4}{5}}={32}^{\frac{4}{5}}=2^4=16$   

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d)

${\left(16a^8{b}^4\right)}^{-\frac{3}{4}}={\left(\frac{1}{16a^8{b}^4}\right)}^{\frac{3}{4}}=\frac{1}{{16}^{\frac{3}{4}}{a}^{8\cdot \frac{3}{4}}{b}^{4\cdot \frac{3}{4}}}=\frac{1}{2^3{a}^6{b}^3}=\frac{1}{8}{a}^{-6}{b}^{-3}$

${\left(a^{-\frac{5}{3}}{b}^{\frac{5}{4}}\right)}^{\frac{12}{5}}=a^{-\frac{5}{3}\cdot \frac{12}{5}}{b}^{\frac{5}{4}\cdot \frac{12}{5}}=a^{-4}{b}^3$

(Uwaga: błąd w odpowiedziach.)

${\left(x^{\frac{m}{n}}{y}^{\frac{p}{q}}\right)}^{-\frac{nq}{mp}}=x^{\frac{m}{n}\cdot \left(-\frac{nq}{mp}\right)}{y}^{\frac{p}{q}\cdot \left(-\frac{nq}{mp}\right)}=x^{-\frac{q}{p}}{y}^{-\frac{n}{m}}$    

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e)

${\left(7\frac{1}{2}\right)}^{\frac{2}{3}}\cdot {\left(2\frac{2}{3}\right)}^{\frac{2}{3}}\cdot {\left(3\frac{1}{5}\right)}^{\frac{2}{3}}={\left(\frac{\cancel{15}}{2}\cdot \frac{8}{\cancel{3}}\cdot \frac{16}{\cancel{5}}\right)}^{\frac{2}{3}}={\left(4\cdot 16\right)}^{\frac{2}{3}}={64}^{\frac{2}{3}}=4^2=16$  

${\left(2\frac{2}{5}\right)}^{\frac{3}{5}}\cdot {\left(6\frac{2}{3}\right)}^{\frac{3}{5}}\cdot {\left(\frac{1}{16}\right)}^{\frac{3}{5}}={\left(\frac{{\cancel{12}}^4}{\cancel{5}}\cdot \frac{{\cancel{20}}^4}{\cancel{3}}\cdot \frac{1}{16}\right)}^{\frac{3}{5}}={\left(\frac{4\cdot 4}{16}\right)}^{\frac{3}{5}}=1^{\frac{3}{5}}=1$  

${\left(4\frac{7}{9}\right)}^{\frac{1}{2}}\cdot {\left(2\frac{1}{4}\right)}^{\frac{1}{2}}\cdot {\left(10\frac{3}{4}\right)}^{\frac{1}{2}}={\left(\frac{43}{\cancel{9}}\cdot \frac{\cancel{9}}{4}\cdot \frac{43}{4}\right)}^{\frac{1}{2}}={\left(\frac{{43}^2}{4^2}\right)}^{\frac{1}{2}}=\frac{43}{4}=10\frac{3}{4}$