[tex]a) \ 4^{6}\cdot0,5^{12}= 4^{6}\cdot(0,5^{2})^{6}=4^{6}\cdot(0,25)^{6} =(4\cdot0,25)^{6} =1^{6} = 1[/tex]
[tex]b) \ 0,5^{9}\cdot800^{3} = (\frac{1}{2})^{9}\cdot800^{3} = ((\frac{1}{2})^{3})^{3}\cdot800^{3} = (\frac{1}{8})^{3}\cdot800^{3} = (\frac{1}{8}\cdot800)^{3} = (\frac{800}{8})^{3}} =\\\\= 100^{3} =(10^{2})^{3} = 10^{6} = 1 \ 000 \ 000[/tex]
[tex]c) \ 8^{2}\cdot2^{12}\cdot0,125^{6} = (2^{3})^{2}\cdot(2^{2})^{6}\cdot(0,125)^{6} =2^{6}\cdot4^{6}\cdot0,125^{6} = (2\cdot4\cdot0,125)^{6} = 1^{6} = 1[/tex]
[tex]d) \ 27^{5}:(3^{4}\cdot9^{2})^{2} = (3^{3})^{5}:(3^{4}\cdot(3^{2})^{2})^{2} =3^{15}:(3^{4}\cdot3^{4})^{2} = 3^{15}:(3^{8})^{2} = \\\\=3^{15}:3^{16} = 3^{15-16}=3^{-1} = \frac{1}{3}[/tex]
[tex]e) \ (\frac{1}{4})^{5}:0,125^{3} = ((\frac{1}{2})^{2})^{5}:(\frac{125}{100})^{3} = (\frac{1}{2})^{10}:(\frac{1}{8})^{3} = (\frac{1}{2})^{10}:((\frac{1}{2})^{3})^{3} =\\\\= (\frac{1}{2})^{10}:(\frac{1}{2})^{9}=(\frac{1}{2})^{10-9} = (\frac{1}{2})^{1} = \frac{1}{2}[/tex]
[tex]f) \ 0,027^{4}\cdot0,09^{2}\cdot0,3:0,0081^{4} =(\frac{27}{1000})^{4}\cdot(\frac{9}{100})^{2}\cdot\frac{3}{10}:(\frac{81}{10000})^{4}=\\\\=((\frac{3}{10})^{3})^{4}\cdot((\frac{3}{10})^{2})^{2}\cdot\frac{3}{10}:((\frac{3}{10})^{4})^{4} = (\frac{3}{10})^{12}\cdot(\frac{3}{10})^{4}\cdot\frac{3}{10}:(\frac{3}{10})^{16}=(\frac{3}{10})^{17}:(\frac{3}{10})^{16}=\\\\=(\frac{3}{10})^{17-16} = (\frac{3}{10})^{1}=\frac{1}{3}[/tex]
Wykorzystano wzory:
[tex]a^{m}\cdot a^{n} = a^{m+n}\\\\a^{m}:a^{n} = a^{m-n}\\\\(a^{m})^{n} = a^{m\cdot n}\\\\a^{n}\cdot b^{n} = (a\cdot b)^{n}\\\\a^{n}:b^{n} = (\frac{a}{b})^{n}[/tex]
[tex]Dla \ a,b \in (0,+\infty)[/tex]
