\[\boxed{\mathbf{914}.}\]
\[1)\ 5x^{\log_{3}2} + 2^{\log_{3}x} = 24\]
\[5 \cdot 2^{\log_{3}x} + 2^{\log_{3}x} = 24\]
\[2^{\log_{3}x}(5 + 1) = 24\]
\[2^{\log_{3}x} \cdot 6 = 24\]
\[2^{\log_{3}x} = 4\]
\[2^{\log_{3}x} = 2^{2}\]
\[\log_{3}x = 2\]
\[3^{2} = x\]
\[x = 9.\]
\[Ответ:x = 9.\]
\[2)\ x^{3\lg^{3}x - \frac{2}{3}\lg x} = 100\sqrt[3]{10};\ \ \]
\[\ x > 0;\ \ \ x \neq 1\]
\[x^{3\lg^{3}x - \frac{2}{3}\lg x} = 10^{\frac{7}{3}}\]
\[\lg\left( x^{3\lg^{3}x - \frac{2}{3}\lg x} \right) = \lg\left( 10^{\frac{7}{3}} \right)\]
\[\left( 3\lg^{3}x - \frac{2}{3}\lg x \right)\lg x = \frac{7}{3}\lg 10\]
\[\left( 3\lg^{3}x - \frac{2}{3}\lg x \right)\lg x = \frac{7}{3}\]
\[t = \lg x:\]
\[\left( 3t^{3} - \frac{2}{3}t \right) \cdot t = \frac{7}{3}\]
\[3t^{4} - \frac{2}{3}t^{2} - \frac{7}{3} = 0\ \ \ \ \ \ \ \ \ \ \ | \cdot 3\]
\[9t^{4} - 2t^{2} - 7 = 0\]
\[t^{2} = a \geq 0:\]
\[9a^{2} - 2a - 7 = 0\]
\[D_{1} = 1 + 63 = 64\]
\[a_{1} = \frac{1 + 8}{9} = 1;\ \ \ \ \]
\[\ a_{2} = \frac{1 - 8}{9} =\]
\[= - \frac{7}{9}\ (не\ подходит).\]
\[t^{2} = 1\]
\[t = \pm 1.\]
\[\textbf{а)}\ \lg x = 1\]
\[x = 10^{1} = 10.\ \ \]
\[\textbf{б)}\ \lg x = - 1\]
\[x = 10^{- 1} = \frac{1}{10}.\]
\[Ответ:x = 1;\ \ x = \frac{1}{10}\text{.\ }\]