7.
[tex]\dfrac{25^{1005}}{5^{2011}-5^{2010}} = \dfrac{(5^{2})^{1005}}{5^{2010}\cdot5-5^{2010}} = \dfrac{5^{2010}}{5^{2010}(5-1)} = \dfrac{1}{4}[/tex]
9.
[tex]\dfrac{(12^{\frac{5}{6}})^{\frac{1}{5}}\cdot(\frac{1}{4})^{\frac{1}{6}}}{(3^{-2\frac{1}{2}})^{\frac{1}{3}}}=\dfrac{12^{\frac{1}{6}}\cdot(\frac{1}{4})^{\frac{1}{6}}}{(3^{-\frac{5}{2}})^{\frac{1}{3}}}=\dfrac{(12\cdot\frac{1}{4})^{\frac{1}{6}}}{3^{-\frac{5}{6}}}=\dfrac{3^{\frac{1}{6}}}{3^{-\frac{5}{6}}}=3^{\frac{1}{6}-(-\frac{5}{6})}=3^{1} = 3[/tex]
10.
[tex]lo_{36}\dfrac{log_232+log_381}{log_{16} 4\cdot log_{4}2}=log_{36}\dfrac{log_{2}2^{5}+log_3 3^{4}}{log_{16}16^{\frac{1}{2}}\cdot log_4 4^{\frac{1}{2}}}=log_{36}\dfrac{5+4}{\frac{1}{2}\cdot\frac{1}{2}}=log_{36}\dfrac{9}{\frac{1}{4}}=\\\\\\=log_{36}(9\cdot4) = log_{36} 36 =log_{36}36^{1} = 1[/tex]
Wykorzystano wzory:
[tex](a^{m})^{n} = a^{m\cdot n}\\\\a^{m}\cdot a^{n} = a^{m+n}\\\\a^{m}:a^{n} = a^{m-n}[/tex]
