a)   ${10}^8:{10}^{-3}={10}^{8-\left(-3\right)}={10}^{8+3}={10}^{11}$  

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b)   ${10}^{12}\cdot {\left({10}^2\right)}^{-2}={10}^{12}\cdot {10}^{2\cdot \left(-2\right)}={10}^{12}\cdot {10}^{-4}={10}^{12+\left(-4\right)}={10}^{12-4}={10}^8$  

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c)   ${\left(7^3\right)}^5:7^3=7^{3\cdot 5}:7^3=7^{15}:7^3=7^{15-3}=7^{12}$  

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d)   ${\left(\frac{2}{3}\right)}^5\cdot {\left(\frac{3}{2}\right)}^{-1}={\left(\frac{2}{3}\right)}^5\cdot {\left(\frac{2}{3}\right)}^1={\left(\frac{2}{3}\right)}^{5+1}={\left(\frac{2}{3}\right)}^6$  

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e)   $2^{13}:8^{-3}=2^{13}:{\left(2^3\right)}^{-3}=2^{13}:2^{3\cdot \left(-3\right)}=2^{13}:2^{-9}=2^{13-\left(-9\right)}=2^{13+9}=2^{22}$  

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f)   ${\left(\frac{1}{3}\right)}^8\cdot 9^{-5}\cdot {27}^2=3^{-8}\cdot {\left(3^2\right)}^{-5}\cdot {\left(3^3\right)}^2=3^{-8}\cdot 3^{2\cdot \left(-5\right)}\cdot 3^{3\cdot 2}=3^{-8}\cdot 3^{-10}\cdot 3^6=$  

$=3^{-8+\left(-10\right)+6}=3^{-8-10+6}=3^{-12}$