[tex]1) \: {3}^{5} \times ( { - \frac{1}{6}) }^{5} \div ( - { \frac{1}{2}) }^{4} + (3 \frac{1}{2} - {2}^{2} ) {}^{2} = \\ \\ - {3}^{5} \times ( \frac{1}{6} ) {}^{5} \div ( \frac{1}{2} ) {}^{4} + ( \frac{7}{2} - 4) {}^{2} = \\ \\ - {3}^{5} \times \frac{1}{7776} \times \frac{1}{16} + ( - \frac{1}{2} ) {}^{2} = \\ \\ - {3}^{5} \times \frac{1}{7776} \times 16 + \frac{1}{4} = - {3}^{5} \times \frac{1}{486} + \frac{1}{4} = \\ \\ - \frac{ {3}^{5} }{486} + \frac{1}{4} = - \frac{243}{486} + \frac{1}{4} = \\ \\ - \frac{1}{2} + \frac{1}{4 } = \frac{ - 2 + 1}{4} = - \frac{ 1}{4} [/tex]
[tex]2) \: (3.4 \times {10}^{17} ) \times (4 \times {10}^{5} ) = \\ \\ (3.4 \times 4) \times ( {10}^{17} \times {10}^{5} ) = \\ \\ 13.6 \times {10}^{22} = 1.36 \times {10}^{23} [/tex]
[tex] \\ 3) \: \frac{ {5}^{3} \times {5}^{8} \times 4- {5}^{3} \times {5}^{6} }{6 \times ( {5}^{22} \times {5}^{11}) } = \\ \\ \frac{ {5}^{3} \times {5}^{3} \times {5}^{3} \times ( {5}^{2} \times 4 - 1)}{6 \times {5}^{33} } = \\ \\ \frac{25 \times 4 - 1}{6 \times {5}^{24} } = \frac{100 - 1}{6 \times {5}^{24} } = \\ \\ \frac{99}{6 \times {5}^{24} } = \frac{33}{2 \times {5}^{24} } [/tex]
[tex] \\ 4) \: \frac{ {2}^{12} \times {30}^{13} \times {10}^{7} }{5 \times {6}^{12} \times {25}^{9} \times {8}^{6} } 3 \\ \\ \frac{ {2}^{12} \times {5}^{13} \times {6}^{13} \times {5}^{7} \times {2}^{7} }{5 \times {6}^{12} \times {5}^{18} \times {2}^{18} } = \\ \\ \frac{ {5}^{12} \times 6 \times {5}^{7} \times {2}^{7} }{ {5}^{18} \times {2}^{6} } = \\ \\ \frac{6 \times {5}^{7} \times 2}{ {5}^{6} } = \\ \\ 6 \times 5 \times 2 = 60[/tex]
