Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 786

\[\boxed{\mathbf{786}\mathbf{.}}\]

\[1)\ 36^{\log_{6}5} + 10^{1 - \log_{10}2} -\]

\[- 8^{\log_{2}3} = \left( 6^{\log_{6}5} \right)^{2} + \frac{10}{10^{\log_{10}2}} -\]

\[- \left( 2^{\log_{2}3} \right)^{3} =\]

\[= 5^{2} + \frac{10}{2} - 3^{3} =\]

\[= 25 + 5 - 27 = 3\]

\[2)\ \left( 81^{\frac{1}{4} - \frac{1}{2}\log_{9}4} + 25^{\log_{125}8} \right) \bullet\]

\[\bullet 49^{\log_{7}2} =\]

\[= \left( \frac{81^{\frac{1}{4}}}{81^{\frac{1}{2}\log_{9}4}} + 5^{2\log_{125}8} \right) \bullet\]

\[\bullet \left( 7^{\log_{7}2} \right)^{2} =\]

\[= \left( \frac{3}{9^{\log_{9}4}} + \left( 125^{\log_{125}8} \right)^{\frac{2}{3}} \right) \bullet 2^{2} =\]

\[= \left( \frac{3}{4} + 8^{\frac{2}{3}} \right) \bullet 2^{2} =\]

\[= \left( \frac{3}{2^{2}} + \left( 2^{3} \right)^{\frac{2}{3}} \right) \bullet 2^{2} =\]

\[= \left( \frac{3}{2^{2}} + 2^{2} \right) \bullet 2^{2} = 3 + 2^{4} =\]

\[= 3 + 16 = 19\]

\[3)\ 16^{1 + \log_{4}5} + 4^{\frac{1}{2}\log_{2}3 + 3\log_{8}5} =\]

\[= 16 \bullet 16^{\log_{4}5} +\]

\[+ 4^{\frac{1}{2}\log_{2}3} \bullet 4^{3\log_{8}5} =\]

\[= 16 \bullet \left( 4^{\log_{4}5} \right)^{2} +\]

\[+ 2^{\log_{2}3} \bullet 64^{\log_{8}5} = 16 \bullet 5^{2} +\]

\[+ 3 \bullet \left( 8^{\log_{8}5} \right)^{2} =\]

\[= 16 \bullet 25 + 3 \bullet 5^{2} =\]

\[= 400 + 3 \bullet 25 =\]

\[= 400 + 75 = 475\]