Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 704

\[1)\ 2^{999}:\]

\[2^{999} = \left( \left( 2^{9} \right)^{3} \right)^{37} = \left( (512)^{3} \right)^{37} =\]

\[= \left( 12^{3} \right)^{37} = 1728^{37} = 28^{37} =\]

\[= 28 \bullet \left( 28^{3} \right)^{12} =\]

\[= 28 \bullet 21\ 952^{12} = 28 \bullet 52^{12} =\]

\[= 28 \bullet \left( 52^{3} \right)^{4} = 28 \bullet 140\ 608^{4} =\]

\[= 28 \bullet 8^{4} = 28 \bullet 4096 =\]

\[= 28 \bullet 96 = 2688 = 88.\]

\[Ответ:\ \ 88.\]

\[2)\ 3^{999}:\]

\[3^{999} = \left( \left( 2^{9} \right)^{3} \right)^{37} = \left( (19\ 683)^{3} \right)^{37} =\]

\[= \left( 83^{3} \right)^{37} = {571\ 787}^{37} =\]

\[= 87^{37} = 87 \bullet \left( 87^{3} \right)^{12} =\]

\[= 87 \bullet 658\ 503^{12} = 87 \bullet 3^{12} =\]

\[= 87 \bullet \left( 3^{3} \right)^{4} = 87 \bullet 27^{4} =\]

\[= 87 \bullet 531\ 441 = 87 \bullet 41 =\]

\[= 3567 = 67.\]

\[Ответ:\ \ 67.\]