a)   $2^{82}:4^{40}=2^{82}:{\left(2^2\right)}^{40}=2^{82}:2^{2\cdot 40}=2^{82}:2^{80}=2^{82-80}=2^2=4$    

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b)   $\frac{{18}^4}{9^4}={\left(\frac{18}{9}\right)}^4=2^4=16$  

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c)   $\frac{2^{15}\cdot {27}^2}{6^{15}}=\frac{2^{15}\cdot {\left(3^3\right)}^2}{{\left(2\cdot 3\right)}^{15}}=\frac{2^{15}\cdot 3^{3\cdot 2}}{2^{15}\cdot 3^{15}}=\frac{\cancel{2^{15}}\cdot 3^6}{\cancel{2^{15}}\cdot 3^{15}}=\frac{3^6}{3^{15}}=3^{6-15}=3^{-9}$  

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d)   ${\left(5^2\right)}^3\cdot 2^6=5^{2\cdot 3}\cdot 2^6=5^6\cdot 2^6={\left(5\cdot 2\right)}^6={10}^6=1000000$  

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e)   ${0,2}^8\cdot {25}^4={\left(\frac{1}{5}\right)}^8\cdot {\left(5^2\right)}^4=5^{-8}\cdot 5^{2\cdot 4}=5^{-8}\cdot 5^8=5^{-8+8}=5^0=1$  

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f)   ${0,01}^{12}\cdot {100}^3={\left(\frac{1}{100}\right)}^{12}\cdot {\left({10}^2\right)}^3={100}^{-12}\cdot {10}^{2\cdot 3}$  

$={\left({10}^2\right)}^{-12}\cdot {10}^{2\cdot 3}={10}^{2\cdot \left(-12\right)}\cdot {10}^6={10}^{-24}\cdot {10}^6={10}^{-24+6}={10}^{-18}$  

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g)   ${25}^3\cdot {0,5}^{-6}={\left(5^2\right)}^3\cdot {\left(\frac{1}{2}\right)}^{-6}=5^{2\cdot 3}\cdot 2^6=5^6\cdot 2^6={\left(5\cdot 2\right)}^6={10}^6=1000000$  

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h)   $5^8:{\left(0,2\right)}^{-7}=5^8:{\left(\frac{1}{5}\right)}^{-7}=5^8:5^7=5^{8-7}=5^1=5$  

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i)   $\frac{{125}^3\cdot {\left(5^{-2}\right)}^4}{{25}^4\cdot {25}^{-5}}=\frac{{\left(5^3\right)}^3\cdot 5^{-2\cdot 4}}{{\left(5^2\right)}^4\cdot {\left(5^2\right)}^{-5}}=\frac{5^{3\cdot 3}\cdot 5^{-8}}{5^{2\cdot 4}\cdot 5^{2\cdot \left(-5\right)}}=\frac{5^9\cdot 5^{-8}}{5^8\cdot 5^{-10}}=\frac{5^{9+\left(-8\right)}}{5^{8+\left(-10\right)}}=\frac{5^1}{5^{8-10}}=$  

$=\frac{5^1}{5^{-2}}=5^{1-\left(-2\right)}=5^{1+2}=5^3=125$  

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j)   $\frac{6^{-3}\cdot {\left(6^2\right)}^{-5}}{{\left(\frac{1}{36}\right)}^6}=\frac{6^{-3}\cdot 6^{2\cdot \left(-5\right)}}{{\left(\frac{1}{6^2}\right)}^6}=\frac{6^{-3}\cdot 6^{-10}}{{\left({\left(\frac{1}{6}\right)}^2\right)}^6}=\frac{6^{-3+\left(-10\right)}}{{\left(\frac{1}{6}\right)}^{2\cdot 6}}=\frac{6^{-3-10}}{{\left(\frac{1}{6}\right)}^{12}}=\frac{6^{-13}}{6^{-12}}=$  

$=6^{-13-\left(-12\right)}=6^{-13+12}=6^{-1}=\frac{1}{6}$  

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k)   $\frac{{\left({121}^2\right)}^3}{{\left(\frac{1}{11}\right)}^{-8}}\cdot {11}^{-2}=\frac{{121}^{2\cdot 3}}{{11}^8}\cdot {11}^{-2}=\frac{{121}^6}{{11}^8}\cdot {11}^{-2}=\frac{{\left({\left(11\right)}^2\right)}^6}{{11}^8}\cdot {11}^{-2}=\frac{{11}^{2\cdot 6}}{{11}^8}\cdot {11}^{-2}=$  

$=\frac{{11}^{12}}{{11}^8}\cdot {11}^{-2}={11}^{12-8}\cdot {11}^{-2}={11}^4\cdot {11}^{-2}=1^{4+\left(-2\right)}={11}^{4-2}={11}^2=121$  

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l)   $\frac{7^5:49}{7}\cdot {\left(\frac{1}{7}\right)}^{-2}=\frac{7^5:7^2}{7^1}\cdot 7^2=\frac{7^{5-2}}{7^1}\cdot 7^2=\frac{7^3}{7^1}\cdot 7^2=7^{3-1}\cdot 7^2=7^2\cdot 7^2=7^{2+2}=7^4$