$\text{a)}{1000}^4={\left({10}^3\right)}^4={10}^{3\cdot 4}={10}^{12}$  

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$\text{b)}{\left(0,01\right)}^5={\left(\frac{1}{100}\right)}^5={\left(\frac{1}{{10}^2}\right)}^5={\left({\left(\frac{1}{10}\right)}^2\right)}^5={\left(\frac{1}{10}\right)}^{2\cdot 5}={\left(\frac{1}{10}\right)}^{10}={10}^{-10}$  

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$\text{c)}{100}^{-4}={\left({10}^2\right)}^{-4}={10}^{2\cdot \left(-4\right)}={10}^{-8}$  

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$\text{d)}{1000}^{-3}\cdot {0,1}^8={\left({10}^3\right)}^{-3}\cdot {\left(\frac{1}{10}\right)}^8={10}^{3\cdot \left(-3\right)}\cdot {10}^{-8}={10}^{-9}\cdot {10}^{-8}={10}^{\left(-9\right)+\left(-8\right)}=$  

$={10}^{-9-8}={10}^{-17}$  

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$\text{e)}{100}^{-7}:{0,001}^9={\left({10}^2\right)}^{-7}:{\left(\frac{1}{1000}\right)}^9={10}^{2\cdot \left(-7\right)}:{\left(\frac{1}{{10}^3}\right)}^9={10}^{2\cdot \left(-7\right)}:{\left({\left(\frac{1}{10}\right)}^3\right)}^9=$  

$={10}^{-14}:{\left(\frac{1}{10}\right)}^{3\cdot 9}={10}^{-14}:{\left(\frac{1}{10}\right)}^{27}={10}^{-14}:{10}^{-27}={10}^{-14-\left(-27\right)}=$  

$={10}^{-14+27}={10}^{13}$  

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$\text{f)}{0,1}^5:{0,01}^6={\left(\frac{1}{10}\right)}^5:{\left(\frac{1}{100}\right)}^6={10}^{-5}:{\left(\frac{1}{{10}^2}\right)}^6={10}^{-5}:{\left({\left(\frac{1}{10}\right)}^2\right)}^6=$  

$={10}^{-5}:{\left(\frac{1}{10}\right)}^{12}={10}^{-5}:{10}^{-12}={10}^{-5-\left(-12\right)}={10}^{-5+12}={10}^7$  

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$\text{g)}{0,001}^{-2}\cdot {0,01}^{-3}\cdot {0,1}^{-4}={\left(\frac{1}{1000}\right)}^{-2}\cdot {\left(\frac{1}{100}\right)}^{-3}\cdot {\left(\frac{1}{10}\right)}^{-4}=$  

$={\left(\frac{1}{{10}^3}\right)}^{-2}\cdot {\left(\frac{1}{{10}^2}\right)}^{-3}\cdot {\left(\frac{1}{10}\right)}^{-4}={\left({\left(\frac{1}{10}\right)}^3\right)}^{-2}\cdot {\left({\left(\frac{1}{10}\right)}^2\right)}^{-3}\cdot {10}^4=$  

$={\left(\frac{1}{10}\right)}^{3\cdot \left(-2\right)}\cdot {\left(\frac{1}{10}\right)}^{2\cdot \left(-3\right)}\cdot {10}^4={\left(\frac{1}{10}\right)}^{-6}\cdot {\left(\frac{1}{10}\right)}^{-6}\cdot {10}^4=$  

$={10}^6\cdot {10}^6\cdot {10}^4={10}^{6+6+4}={10}^{16}$  

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$\text{h)}{0,1}^4\cdot {100}^{-12}:{1000}^{-5}={\left(\frac{1}{10}\right)}^4\cdot {\left({10}^2\right)}^{-12}:{\left({10}^3\right)}^{-5}={10}^{-4}\cdot {10}^{2\cdot \left(-12\right)}:{10}^{3\cdot \left(-5\right)}=$  

$={10}^{-4}\cdot {10}^{-24}:{10}^{-15}={10}^{-4+\left(-24\right)-\left(-15\right)}={10}^{-4-24+15}={10}^{-13}$  

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$\text{i)}{\left({0,01}^{-3}\right)}^{-5}\cdot {\left({100}^2\right)}^3={\left({\left(\frac{1}{100}\right)}^{-3}\right)}^{-5}\cdot {100}^{2\cdot 3}={\left(\frac{1}{100}\right)}^{\left(-3\right)\cdot \left(-5\right)}\cdot {100}^6=$  

$={\left(\frac{1}{100}\right)}^{15}\cdot {\left({10}^2\right)}^6={\left(\frac{1}{{10}^2}\right)}^{15}\cdot {10}^{2\cdot 6}={\left(\frac{1}{10}\right)}^{2\cdot 15}\cdot {10}^{12}={\left(\frac{1}{10}\right)}^{30}\cdot {10}^{12}=$  

$={10}^{-30}\cdot {10}^{12}={10}^{-30+12}={10}^{-18}$