[tex]a) \ \dfrac{2^{5}\cdot4^{3}\cdot8}{16\cdot2^{8}} = \dfrac{2^{5}\cdot(2^{2})^{3}\cdot2^{3}}{2^{4}\cdot2^{8}} =\dfrac{2^{5}\cdot2^{2\cdot3}\cdot2^{3}}{2^{4+8}} = \dfrac{2^{5+6+3}}{2^{12}} = \dfrac{2^{14}}{2^{12}} = 2^{14-12} = 2^{2}[/tex]
[tex]b) \ \dfrac{3^{8}:27\cdot9^{4}}{3^{2}\cdot81} = \dfrac{3^{8}:3^{3}\cdot(3^{2})^{4}}{3^{2}\cdot3^{4}} = \dfrac{3^{8}:3^{3}\cdot3^{2\cdot4}}{3^{2+4}} = \dfrac{3^{8}:3^{3}\cdot3^{8}}{3^{6}} =\dfrac{3^{8-3+8}}{3^{6}} = \dfrac{3^{13}}{3^{6}} = \\\\=3^{13-6} = 3^{7}[/tex]
[tex]c) \ (0,01)^{2}\cdot(0,1)^{4}:(0,001)=((0,1)^{2})^{2}\cdot(0,1)^{4}:(0,1)^{3} = \\\\=(0,1)^{4}\cdot(0,1)^{4}:(0,1)^{3} =(0,1)^{8}:(0,1)^{3} = (0,1)^{5}[/tex]
[tex]d) \ (\dfrac{3}{2})^{4}\cdot(\dfrac{9}{4})^{3}:\dfrac{27}{8} = (\dfrac{3}{4})^{4}\cdot((\dfrac{3}{2})^{2})^{3}:(\dfrac{3}{2})^{3} = (\dfrac{3}{2})^{4}\cdot(\dfrac{3}{2})^{6}:(\dfrac{3}{2})^{3} =(\dfrac{3}{2})^{10}:(\dfrac{3}{2})^{3} =\\\\= (\dfrac{3}{2})^{7}[/tex]
Wyjaśnienie
Wykorzystano prawa potęg:
[tex](a^{m})^{n} = a^{m\cdot n}\\\\a^{m}\cdot a^{n} =a^{m+n}\\\\a^{m}:a^{n} = a^{m-n}[/tex]
