4.
[tex]\frac{2^{10}:(2^{2}\cdot2^{4})}{2^{3}} = \frac{2^{10}:2^{6}}{2^{3}} = \frac{2^{4}}{2^{3}} = 2^{1} = 2[/tex]
5.
[tex]a) \ (6^{9})^{8} = 6^{72}\\\\b) \ ((-5)^{7})^{3} = (-5)^{21}\\\\c) \ ((-0,1)^{6})^{5} = (-0,1)^{30} = 0,1^{30}\\\\d) \ ((3^{2})^{3})^{6} = 3^{36}[/tex]
6)
[tex]a) \ 2^{9}\cdot5^{7} = 2^{2}\cdot2^{7}\cdot5^{7} = 4\cdot(2\cdot5)^{7} = 4\cdot10^{7} = 40 \ 000 \ 000\\\\b) \ 0,2^{6}\cdot5^{3} = 0,2^{3}\cdot0,2^{3}\cdot5^{3} = (0,2\cdot0,2\cdot5)^{3} = (0,2\cdot1)^{3} = 0,2^{3} = 0,008\\\\c) \ (\frac{2}{3})^{6}:(\frac{4}{9})^{5} =(\frac{2}{3})^{6}:((\frac{2}{3})^{2})^{5} =(\frac{2}{3})^{6}:(\frac{2}{3})^{10} = (\frac{2}{3})^{-4} = (\frac{3}{2})^{4} = \frac{3^{4}}{2^{4}} = \frac{81}{16} = 5\frac{1}{16}[/tex]
Wykorzystano własności potęg:
[tex]a^{m}\cdot a^{n} = a^{m+n}\\\\(a^{m})^{n} = a^{m\cdot n}\\\\a^{m}:a^{n}=a^{m-n}[/tex]
[tex](\frac{a}{b})^{-n} = (\frac{b}{a})^{n}[/tex]
