Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 216

\[\boxed{\mathbf{216}\mathbf{.}}\]

\[1)\ 10^{x} = \sqrt[3]{100}\]

\[10^{x} = \sqrt[3]{10^{2}}\]

\[10^{x} = 10^{\frac{2}{3}}\]

\[Ответ:\ \ x = \frac{2}{3}.\]

\[2)\ 10^{x} = \sqrt[5]{10\ 000}\]

\[10^{x} = \sqrt[5]{10^{4}}\]

\[10^{x} = 10^{\frac{4}{5}}\]

\[x = \frac{4}{5} = 0,8\]

\[Ответ:\ \ x = 0,8.\]

\[3)\ 225^{2x^{2} - 24} = 15\]

\[15^{2\left( 2x^{2} - 24 \right)} = 15^{1}\]

\[2\left( 2x^{2} - 24 \right) = 1\]

\[4x^{2} - 48 = 1\]

\[4x^{2} = 49\]

\[x^{2} = \frac{49}{4}\]

\[x = \pm \sqrt{\frac{49}{4}} = \pm \frac{7}{2} = \pm 3,5.\]

\[Ответ:\ \ x = \pm 3,5.\]

\[4)\ 10^{x} = \frac{1}{\sqrt[4]{10\ 000}}\]

\[10^{x} = \frac{1}{\sqrt[4]{10^{4}}}\]

\[10^{x} = \frac{1}{10}\]

\[10^{x} = 10^{- 1}\]

\[Ответ:\ \ x = - 1.\]

\[5)\ \left( \sqrt{10} \right)^{x} = 10^{x^{2} - x}\]

\[\left( 10^{\frac{1}{2}} \right)^{x} = 10^{x^{2} - x}\]

\[10^{\frac{x}{2}} = 10^{x^{2} - x}\]

\[\frac{x}{2} = x^{2} - x\]

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

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

\[x(2x - 3) = 0\]

\[x_{1} = 0;\ \text{\ \ }x_{2} = \frac{3}{2} = 1,5.\]

\[Ответ:\ \ x_{1} = 0;\ \ \ x_{2} = 1,5.\]

\[6)\ 100^{x^{2} - 1} = 10^{1 - 5x}\]

\[10^{2\left( x^{2} - 1 \right)} = 10^{1 - 5x}\]

\[2\left( x^{2} - 1 \right) = 1 - 5x\]

\[2x^{2} - 2 = 1 - 5x\]

\[2x^{2} + 5x - 3 = 0\]

\[D = 5^{2} + 4 \bullet 2 \bullet 3 = 25 + 24 = 4\]

\[x_{1} = \frac{- 5 - 7}{2 \bullet 2} = - \frac{12}{4} = - 3;\ \]

\[x_{2} = \frac{- 5 + 7}{2 \bullet 2} = \frac{2}{4} = 0,5.\]

\[Ответ:\ \ x_{1} = - 3;\ \ \ x_{2} = 0,5.\]