6.
[tex]a) \ \frac{(3^{5})^{3}\cdot3^{2}}{3^{8}\cdot3^{6}} = \frac{3^{5\cdot3}\cdot3^{2}}{3^{8+6}} = \frac{3^{15}\cdot3^{2}}{3^{14}} = \frac{3^{15+2}}{3^{14}} = \frac{3^{17}}{3^{14}} = 3^{17-14} = 3^{3} = 27\\\\\\b) \ (10^{7}:10)^{2}:10^{11} =(10^{6})^{2}:10^{11} = 10^{12}:10^{11} = 10^{1} = 10\\\\\\c) \ ((3^{7}\cdot3^{2})^{2}:3^{17})^{3} = ((3^{9})^{2}:3^{17})^{3} = (3^{18}:3^{17})^{3} = (3^{1})^{3} = 3^{3} = 27[/tex]
[tex]d) \ \frac{10^{5}\cdot10^{3}\cdot(10^{2})^{7}}{10^{20}}=\frac{10^{8}\cdot10^{14}}{10^{20}} = \frac{10^{22}}{10^{20}}=10^{2} = 100\\\\\\e) \ \frac{(7^{6}\cdot7^{4})^{3}}{(7^{12}:7^{8})^{7}} = \frac{(7^{10})^{3}}{(7^{4})^{7}}=\frac{7^{30}}{7^{28}} = 7^{2} = 49[/tex]
[tex]f) \ 7^{5}\cdot(7^{9}:7^{6})^{2} :(7^{2})^{5} = 7^{5}\cdot(7^{3})^{2}:7^{10} = 7^{5}\cdot7^{6}:7^{10} = 7^{11}:7^{10} = 7^{1} = 7[/tex]
Wykorzystane wzory:
[tex]a^{m}\cdot a^{n} = a^{m+n} \ - \ iloczyn \ poteg\\\\a^{m}:a^{n} = a^{m-n} \ - \ iloraz \ poteg\\\\(a^{m})^{n} = a^{m\cdot n} \ - \ potega \ potegi[/tex]
