Задание 2.8

Question

Решите уравнение:
\begin{align*}
1) &\frac{\sqrt{32}}{16^{x^{2}}}=8^{3 x}; &\quad 3) 2^{x-1}=12^{2 x} \cdot 3^{-2 x} \cdot 2^{x+1};\
2) &9 \cdot 3^{\sin x}=\sqrt{27};& 4) \sqrt[5]{7^{x+1}}=\frac{49}{\sqrt{7}}
\end{align*}

Gimli · Accepted Answer

\begin{enumerate}
\item $$\frac{\sqrt{32}}{16^{x^2}} = 8^{3x}$$
$$\sqrt{32} = 2^{4x^2 + 9x}$$
$$2^{\frac{5}{2}} = 2^{4x^2 + 9x}$$
$$4x^2 + 9x = \frac{5}{2}$$
$$8x^2 + 18x - 5 = 0$$
$$D = 18^2 + 8\cdot 5\cdot 4 = 324 + 160 = 484 = 22^2$$
$$x_1 = \frac{-18 + 22}{16} = \frac{4}{16} = \frac{1}{4}, \quad x_2 = \frac{-40}{16} = -2.5$$
	extbf{Ответ: $x = \{\frac{1}{4}\}$ или $x = \{-2.5\}.$}
\item $$9\cdot 3^{\sin x} = \sqrt{27}$$
$$3^2\cdot 3^{\sin x} = 3^{\frac{3}{2}}$$
$$3^{\sin x +2} = 3^{\frac{3}{2}}$$
$$\sin x + 2 = 1\frac{1}{2}$$
$$\sin x = -\frac{1}{2}$$
$$x = (-1)^n \arcsin \left(-\frac{1}{2}ight) + \pi n$$
$$x = (-1)^{n + 1} \cdot \arcsin \left(\frac{1}{2}ight) + \pi n$$
$$x = (-1)^{n + 1}\cdot \frac{\pi}{6} + \pi n$$
	extbf{Ответ: $x = (-1)^{n + 1}\cdot \frac{\pi}{6} + \pi n$}
\item $$2^{x - 1} = 12^{2x} \cdot 3^{-2x}\cdot 2^{x + 1}$$
$$2^{x - 1} = 2^{4x} \cdot 3^{2x}\cdot 3^{-2x} \cdot 2^{x + 1}$$
$$2^{x - 1} = 2^{5x + 1}$$
$$x - 1 = 5x + 1$$
$$-4x = 2$$
$$x = \{-\frac{1}{2}\}$$
	extbf{Ответ: $x = \{-\frac{1}{2}\}$}
\item $$\sqrt[5]{7^{x + 1}} = \frac{49}{\sqrt{7}}$$
$$\sqrt[5]{7^{x + 1}} = \frac{7^2}{7^{\frac{1}{2}}}$$
$$\sqrt[5]{7^{x + 1}} = 7^{\frac{3}{2}}$$
$$7^{\frac{x+1}{5}} = 7^{\frac{3}{2}}$$
$$\frac{x + 1}{5} = \frac{3}{2}$$
$$2x + 2 = 15$$
$$2x = 13$$
$$x = 6.5$$
	extbf{Ответ: $x = 6.5$}
\end{enumerate}

Задание 2.8

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Задание 2.8

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