$a^n=\underset{n}{\underbrace{a\cdot a\cdot \dots \cdot a}}$	$a^0=1,a\ne 0$	$a^{-n}=\frac{1}{a^n},a\ne 0$
$a^{\frac{m}{n}}=a\sqrt[n]{a^m}$	$a^x\cdot a^y=a^{x+y}$	$a^x:a^y=a^{x-y}$
${\left(a^x\right)}^y=a^{x\cdot y}$	${\left(a\cdot b\right)}^x=a^x\cdot b^x$	${\left(\frac{a}{b}\right)}^x=\frac{a^x}{b^x}$

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a)

$2^{\frac{1}{2}}\cdot 2^{\frac{1}{4}}=2^{\frac{1}{2}+\frac{1}{4}}=2^{\frac{3}{4}}$  

$3^{\frac{1}{2}}\cdot 3^{\frac{1}{3}}=3^{\frac{1}{2}+\frac{1}{3}}=3^{\frac{5}{6}}$

$4^{\frac{1}{2}}\cdot 4^{-\frac{1}{4}}=4^{\frac{1}{2}-\frac{1}{4}}=4^{\frac{1}{4}}$

$5^{\frac{1}{2}}\cdot 5^{-\frac{1}{6}}=5^{\frac{1}{2}-\frac{1}{6}}=5^{\frac{1}{3}}$

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b) 

$2\cdot 2^{-\frac{2}{3}}=2^{1-\frac{2}{3}}=2^{\frac{1}{3}}$

$3^{\frac{1}{2}}\cdot \sqrt{3}=3^{\frac{1}{2}}\cdot 3^{\frac{1}{2}}=3^1=3$

$5^{\frac{2}{3}}\cdot \sqrt[3]{5}=5^{\frac{2}{3}}\cdot 5^{\frac{1}{3}}=5^1=5$       

$6^{\frac{1}{5}}\sqrt[3]{6^3}=6^{\frac{1}{5}}\cdot 6=6^{\frac{6}{5}}$  

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c) 

$2^{\frac{1}{2}}:2^{\frac{1}{4}}=2^{\frac{1}{2}-\frac{1}{4}}=2^{\frac{1}{4}}$   

$3^{\frac{1}{2}}:3^{\frac{1}{3}}=3^{\frac{1}{2}-\frac{1}{3}}=3^{\frac{1}{6}}$

$5^{\frac{1}{2}}:5^{\frac{1}{6}}=5^{\frac{1}{2}-\frac{1}{6}}=5^{\frac{1}{3}}$

$6^{\frac{1}{3}}:6^{\frac{1}{12}}=6^{\frac{1}{3}-\frac{1}{12}}=6^{\frac{1}{4}}$

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d)

$\frac{2^{\frac{7}{15}}}{2^{\frac{1}{5}}}=2^{\frac{7}{15}-\frac{3}{15}}=2^{\frac{4}{15}}$     

$\frac{3^{\frac{3}{4}}}{3^{\frac{7}{12}}}=3^{\frac{9}{12}-\frac{7}{12}}=3^{\frac{1}{6}}$

$\frac{4^{-\frac{3}{5}}}{4^{\frac{1}{2}}}=4^{-\frac{3}{5}-\frac{1}{2}}=4^{-\frac{6}{10}-\frac{5}{10}}=4^{-\frac{11}{10}}$

$\frac{5^{-0,7}}{5^{-0,3}}=5^{-0,7+0,3}=5^{-0,4}$    

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e) 

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

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f) 

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

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g)

$a^{\frac{1}{m}}\cdot a^{\frac{1}{n}}=a^{\frac{1}{m}+\frac{1}{n}}=a^{\frac{m+n}{mn}}$  

$x^{\frac{1}{a+b}}\cdot x^{\frac{1}{a-b}}\cdot x^{\frac{-2a}{a^2-b^2}}=x^{\frac{1}{a+b}+\frac{1}{a-b}+\frac{-2a}{a^2-b^2}}=x^{\frac{a-b+a+b}{a^2-b^2}-\frac{2a}{a^2-b^2}}=x^{\frac{2a-2a}{a^2-b^2}}=x^0=1$  

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h)

$a^{\frac{2}{3}}{b}^{\frac{3}{4}}\cdot a^{\frac{5}{6}}{b}^{\frac{2}{3}}=a^{\frac{2}{3}+\frac{5}{6}}{b}^{\frac{3}{4}+\frac{2}{3}}=a^{\frac{4}{6}+\frac{5}{6}}{b}^{\frac{9}{12}+\frac{8}{12}}=a^{\frac{9}{6}}{b}^{\frac{17}{12}}=a^{\frac{3}{2}}{b}^{\frac{17}{12}}$  

$a^{\frac{1}{n}}{b}^{\frac{1}{m}}\cdot a^{-\frac{1}{n}}{b}^{\frac{2}{m}}=a^{\frac{1}{n}-\frac{1}{n}}{b}^{\frac{1}{m}+\frac{2}{m}}=a^0{b}^{\frac{3}{m}}=b^{\frac{3}{m}}$