Преобразуйте выражение (5x^-1/3y^-2)^-2*15x^3y

Ответ:

\[\ \left( \frac{5x^{- 1}}{3y^{- 2}} \right)^{- 2} \cdot 15x³y = \left( \frac{5y^{2}}{3x} \right)^{- 2} \cdot 15x³y =\]

\[= \frac{9x² \cdot 15x³y}{25y^{4}} = \frac{27x^{5}}{5y^{3}}\]

\[\frac{4^{- 6} \cdot 16^{- 3}}{64^{- 5}} = \frac{\left( 2^{2} \right)^{- 6} \cdot \left( 2^{4} \right)^{- 3}}{\left( 2^{6} \right)^{- 5}} =\]

\[= \frac{2^{- 12} \cdot 2^{- 12}}{2^{- 30}} = \frac{2^{- 24}}{2^{- 30}} = 2^{6} = 64.\]

\[\left( 2,5 \cdot 10^{7} \right) \cdot \left( 6,2 \cdot 10^{- 10} \right) = 15,5 \cdot 10^{- 3} =\]

\[= 1,55 \cdot 10^{- 2}.\]

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

\[= \left( \frac{1 - yx}{x} \right)\left( \frac{xy - 1}{y} \right)^{- 1} =\]

\[= \frac{1 - yx}{x} \cdot \frac{y}{xy - 1} = \frac{xy - 1}{x} \cdot \frac{y}{xy - 1} = - \frac{y}{x}\]

\[\left\{ \begin{matrix} 3 \cdot (x - 1) - 2 \cdot (1 + x) < 1 \\ 3x - 4 > 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix}\text{\ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} 3x - 3 - 2 - 2x < 1 \\ 3x - 4 > 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix}\text{\ \ \ \ \ } \right.\ \left\{ \begin{matrix} x - 5 < 1 \\ 3x > 4\ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x < 6\ \ \ \\ x > 1\frac{1}{3} \\ \end{matrix} \right.\ \]

\[Ответ:\ \ x \in \left( 1\frac{1}{3};6 \right).\]

\[\left( \sqrt{6} + \sqrt{3} \right) \cdot \sqrt{12} - 2\sqrt{6} \cdot \sqrt{3} =\]

\[= \sqrt{6 \cdot 12} + \sqrt{3 \cdot 12} - 2\sqrt{6 \cdot 3} =\]

\[= \sqrt{72} + \sqrt{36} - 2\sqrt{18} =\]

\(= 6\sqrt{2} + 6 - 6\sqrt{2} = 6\)

\[\left( \frac{6}{y^{2} - 9} + \frac{1}{3 - y} \right) \cdot \frac{y^{2} + 6y + 9}{5} =\]

\[= \left( \frac{6}{(y - 3)(y + 3)} - \frac{1}{y - 3} \right) \cdot \frac{(y + 3)^{2}}{5} =\]

\[= \frac{6 - (y + 3)}{(y - 3)(y + 3)} \cdot \frac{(y + 3)^{2}}{5} =\]

\[= \frac{6 - y - 3}{(y - 5)(y + 3)} \cdot \frac{(y + 3)^{2}}{5} =\]

\[= \frac{3 - y}{(y - 3)(y + 3)} \cdot \frac{(y + 3)^{2}}{5} =\]

\[= - \frac{(y - 3) \cdot (y + 3)^{2}}{(y - 3)(y + 3) \cdot 5} = - \frac{y + 3}{5}\]