1)
I.
[tex]\frac{27^{2}:81^{2}\cdot243}{81^{3}:(27^{2})^{2}} = \frac{(3^{3})^{2}:(3^{4})^{2}\cdot3^{5}}{(3^{4})^{3}:((3^{3})^{2})^{2}} = \frac{3^{6}:3^{8}\cdot3^{5}}{3^{12}:3^{12}} = \frac{3^{-2}\cdot3^{5}}{1} = 3^{3} = 27[/tex]
II.
[tex]\frac{(243^{2}:81^{2})^{2}\cdot243\cdot3}{27^{3}\cdot9^{2}:243} = \frac{((243:81)^{2})^{2}\cdot3^{5}\cdot3}{(3^{3})^{3}\cdot(3^{2})^{2}:3^{5}} = \frac{(3^{2})^{2}\cdot3^{6}}{3^{9}\cdot3^{4}:3^{5}} = \frac{3^{4}\cdot3^{6}}{3^{13}:3^{5}} = \frac{3^{10}}{3^{8}} = 3^{2} = 9[/tex]
III.
[tex]\frac{3^{2}:27^{2}\cdot81}{9^{3}:27^{2}} = \frac{3^{2}:(3^{3})^{2}\cdot3^{4}}{(3^{2})^{3}:(3^{3})^{2}} = \frac{3^{2}:3^{6}\cdot3^{4}}{3^{6}:3^{6}} = \frac{3^{-4}\cdot3^{4}}{1} = 3^{0} = 1[/tex]
2)
[tex]\frac{3^{10}\cdot(3^{8}:81^{5})^{2}\cdot27^{10}}{9^{3}\cdot3^{2}} = \frac{3^{10}\cdot((3^{8}:(3^{4})^{5})^{2}\cdot(3^{3})^{10}}{(3^{2})^{3}\cdot3^{2}} =\frac{3^{10}\cdot(3^{8}:3^{20})^{2}\cdot3^{30}}{3^{6}\cdot3^{2}} =[/tex]
[tex]=\frac{3^{10}\cdot(3^{-12})^{2}\cdot3^{30}}{3^{8}}=\frac{3^{10}\cdot3^{-24}\cdot3^{30}}{3^{8}} = \frac{3^{16}}{3^{8}} = 3^{8}[/tex]
Wyjaśnienie:
[tex]a^{m}\cdot a^{n} = a^{m+n}\\a^{m}:a^{n} = a^{m-n}\\(a^{m})^{n} = a^{m\cdot n[/tex]
