$\text{a)}{2}^{\frac{1}{2}}\cdot 2^{1\frac{1}{2}}\cdot 2^{-1}=2^{\frac{1}{2}+1\frac{1}{2}-1}=2^1$  

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$\text{b)}{2}^{-1}\cdot 4^{-\frac{1}{2}}\cdot 8^{\frac{2}{3}}=2^{-1}\cdot {\left(2^2\right)}^{-\frac{1}{2}}\cdot {\left(2^3\right)}^{\frac{2}{3}}=2^{-1}\cdot 2^{-1}\cdot 2^2=2^{-1-1+2}=2^0$  

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$\text{c)}{16}^{\frac{1}{4}}\cdot 8^{-\frac{3}{2}}\cdot \sqrt{8}={\left(2^4\right)}^{\frac{1}{4}}\cdot {\left(2^3\right)}^{-\frac{3}{2}}\cdot {\left(2^3\right)}^{\frac{1}{2}}=2^1\cdot 3^{-\frac{9}{2}}\cdot 2^{\frac{3}{2}}=2^{1-\frac{9}{2}+\frac{3}{2}}=2^{1-\frac{6}{2}}=2^{1-3}=2^{-2}$  

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$\text{d)}{\left({32}^{-\frac{1}{2}}:{64}^{\frac{3}{4}}\right)}^{\frac{1}{2}}\cdot \sqrt{128}={\left({\left(2^5\right)}^{-\frac{1}{2}}:{\left(2^6\right)}^{\frac{3}{4}}\right)}^{\frac{1}{2}}\cdot {\left(2^7\right)}^{\frac{1}{2}}={\left(2^{-\frac{5}{2}}:2^{\frac{9}{2}}\right)}^{\frac{1}{2}}\cdot 2^{\frac{7}{2}}={\left(2^{-\frac{5}{2}-\frac{9}{2}}\right)}^{\frac{1}{2}}\cdot 2^{\frac{7}{2}}={\left(2^{-\frac{14}{2}}\right)}^{\frac{1}{2}}\cdot 2^{\frac{7}{2}}={\left(2^{-7}\right)}^{\frac{1}{2}}\cdot 2^{\frac{7}{2}}=2^{-\frac{7}{2}}\cdot 2^{\frac{7}{2}}=2^{-\frac{7}{2}+\frac{7}{2}}=2^0$  

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$\text{e)}\frac{3^{1\frac{1}{2}}\cdot {\left(\frac{1}{3}\right)}^{1\frac{1}{2}}}{2^{2\frac{2}{3}}\cdot {\left(\frac{1}{2}\right)}^{2\frac{2}{3}}}=\frac{{\left(3\cdot \frac{1}{3}\right)}^{1\frac{1}{2}}}{{\left(2\cdot \frac{1}{2}\right)}^{2\frac{2}{3}}}=\frac{1^{1\frac{1}{2}}}{1^{2\frac{2}{3}}}=\frac{1}{1}=1=2^0$  

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$\text{f)}\frac{{\left(\frac{1}{2}\right)}^3\cdot 2^{-2}\cdot \sqrt[4]{4}}{{64}^{-\frac{3}{4}}\cdot 0,25}=\frac{{\left(2^{-1}\right)}^3\cdot 2^{-2}\cdot {\left(2^2\right)}^{\frac{1}{4}}}{{\left(2^6\right)}^{-\frac{3}{4}}\cdot \frac{1}{4}}=\frac{2^{-3}\cdot 2^{-2}\cdot 2^{\frac{1}{2}}}{2^{-\frac{9}{2}}\cdot 2^{-2}}=\frac{2^{-3-2+\frac{1}{2}}}{2^{-\frac{9}{2}-2}}=\frac{2^{-4\frac{1}{2}}}{2^{-4\frac{1}{2}-2}}=\frac{2^{-4\frac{1}{2}}}{2^{-6\frac{1}{2}}}=2^{-4\frac{1}{2}-\left(-6\frac{1}{2}\right)}=2^{-4\frac{1}{2}+6\frac{1}{2}}=2^2$