Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 1343

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\[1)\ 3^{x - 7} = 81\]

\[3^{x - 7} = 3^{4}\]

\[x - 7 = 4\]

\[x = 11.\]

\[Ответ:\ \ x = 11.\]

\[2)\ 2^{x^{2} - 5x + 6,5} = \sqrt{2}\]

\[2^{x^{2} - 5x + 6,5} = 2^{0,5}\]

\[x^{2} - 5x + 6,5 = 0,5\]

\[x^{2} - 5x + 6 = 0\]

\[D = 25 - 24 = 1\]

\[x_{1} = \frac{5 - 1}{2} = 2;\]

\[x_{2} = \frac{5 + 1}{2} = 3.\]

\[Ответ:\ \ x_{1} = 2;\ \ x_{2} = 3.\]

\[3)\ \left( \frac{1}{4} \bullet 4^{x} \right)^{x} = 2^{2x + 6}\]

\[\left( 4^{x - 1} \right)^{x} = \left( 4^{0,5} \right)^{2x + 6}\]

\[4^{x^{2} - x} = 4^{x + 3}\]

\[x^{2} - x = x + 3\]

\[x^{2} - 2x - 3 = 0\]

\[D = 4 + 12 = 16\]

\[x_{1} = \frac{2 - 4}{2} = - 1;\]

\[x_{2} = \frac{2 + 4}{2} = 3.\]

\[Ответ:\ \ x_{1} = - 1;\ \ x_{2} = 3.\]