Реши примеры на вычисление значения выражения

Ох, задачек тут много! Давай по порядку. **10.15** 1) $12^{\frac{1}{3}} \cdot 6^{\frac{2}{3}} \cdot (0,5)^{\frac{1}{3}} = (12 \cdot 6^2 \cdot 0,5)^{\frac{1}{3}} = (12 \cdot 36 \cdot 0,5)^{\frac{1}{3}} = (216)^{\frac{1}{3}} = 6$ 2) $25^{1,5} + (0,25)^{-0,5} - 81^{0,75} = 25^{\frac{3}{2}} + (\frac{1}{4})^{-\frac{1}{2}} - 81^{\frac{3}{4}} = (5^2)^{\frac{3}{2}} + 4^{\frac{1}{2}} - (3^4)^{\frac{3}{4}} = 5^3 + 2 - 3^3 = 125 + 2 - 27 = 98 + 2 = 100$ 3) $(\frac{1}{16})^{\frac{3}{4}} + (\frac{1}{8})^{-\frac{2}{3}} \cdot (0,81)^{-0,5} = ((\frac{1}{2})^4)^{\frac{3}{4}} + (2^3)^{\frac{2}{3}} \cdot (\frac{81}{100})^{-\frac{1}{2}} = (\frac{1}{2})^3 + 2^2 \cdot (\frac{100}{81})^{\frac{1}{2}} = \frac{1}{8} + 4 \cdot \frac{10}{9} = \frac{1}{8} + \frac{40}{9} = \frac{9 + 320}{72} = \frac{329}{72}$ 4) $16^{\frac{1}{8}} \cdot 8^{\frac{5}{6}} \cdot 4^{1,5} = (2^4)^{\frac{1}{8}} \cdot (2^3)^{\frac{5}{6}} \cdot (2^2)^{\frac{3}{2}} = 2^{\frac{1}{2}} \cdot 2^{\frac{5}{2}} \cdot 2^3 = 2^{\frac{1}{2} + \frac{5}{2} + 3} = 2^{3+3} = 2^6 = 64$ 5) $\frac{10000^{0,4} \cdot 10^{0,5}}{100^{0,3} \cdot 1000^{\frac{1}{6}}} = \frac{(10^4)^{0,4} \cdot 10^{0,5}}{(10^2)^{0,3} \cdot (10^3)^{\frac{1}{6}}} = \frac{10^{1,6} \cdot 10^{0,5}}{10^{0,6} \cdot 10^{\frac{1}{2}}} = \frac{10^{2,1}}{10^{1,1}} = 10^{2,1 - 1,1} = 10^1 = 10$ 6) $\frac{5^{\frac{3}{2}} \cdot 8^{\frac{1}{2}}}{\frac{1}{9^6}} \cdot \frac{\frac{1}{8^4}}{5^2 \cdot 9^{\frac{1}{3}}} = \frac{5^{\frac{3}{2}} \cdot (2^3)^{\frac{1}{2}}}{(3^2)^{\frac{1}{6}}} \cdot \frac{(2^3)^{\frac{1}{4}}}{5^2 \cdot (3^2)^{\frac{1}{3}}} = \frac{5^{\frac{3}{2}} \cdot 2^{\frac{3}{2}}}{3^{\frac{1}{3}}} \cdot \frac{2^{\frac{3}{4}}}{5^2 \cdot 3^{\frac{2}{3}}} = \frac{5^{\frac{3}{2}} \cdot 2^{\frac{3}{2}} \cdot 2^{\frac{3}{4}}}{3^{\frac{1}{3}} \cdot 5^2 \cdot 3^{\frac{2}{3}}} = \frac{5^{\frac{3}{2}} \cdot 2^{\frac{9}{4}}}{5^2 \cdot 3} = \frac{2^{\frac{9}{4}}}{5^{\frac{1}{2}} \cdot 3} = \frac{2^2 \cdot 2^{\frac{1}{4}}}{3 \cdot 5^{\frac{1}{2}}} = \frac{4 \cdot 2^{\frac{1}{4}}}{3 \cdot 5^{\frac{1}{2}}}$ 7) $(72^{\frac{2}{3}})^{\frac{1}{2}} \cdot 2^{\frac{4}{3}} : 36^{\frac{1}{6}} = (2^3 \cdot 3^2)^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} : (6^2)^{\frac{1}{6}} = 2 \cdot 3^{\frac{2}{3}} \cdot 2^{\frac{4}{3}} : 6^{\frac{1}{3}} = 2 \cdot 2^{\frac{4}{3}} \cdot 3^{\frac{2}{3}} : (2 \cdot 3)^{\frac{1}{3}} = 2^{\frac{7}{3}} \cdot 3^{\frac{2}{3}} : 2^{\frac{1}{3}} : 3^{\frac{1}{3}} = 2^2 \cdot 3^{\frac{1}{3}} = 4 \cdot 3^{\frac{1}{3}}$ 8) $(\frac{3^{\frac{5}{6}} \cdot 7^{\frac{5}{6}}}{21^{-1} \cdot 5^{\frac{1}{3}}})^{-6} = (\frac{3^{\frac{5}{6}} \cdot 7^{\frac{5}{6}}}{3^{-1} \cdot 7^{-1} \cdot 5^{\frac{1}{3}}})^{-6} = (3^{\frac{11}{6}} \cdot 7^{\frac{11}{6}} \cdot 5^{-\frac{1}{3}})^{-6} = 3^{-11} \cdot 7^{-11} \cdot 5^2 = \frac{25}{3^{11} \cdot 7^{11}}$ **10.16** 1) $(343^2)^{\frac{1}{2}} \cdot (\frac{1}{49})^{\frac{3}{8}} = 343 \cdot (\frac{1}{7^2})^{\frac{3}{8}} = 7^3 \cdot (7^{-2})^{\frac{3}{8}} = 7^3 \cdot 7^{-\frac{3}{4}} = 7^{\frac{9}{4}}$ 2) $10^{\frac{1}{4}} \cdot 40^{\frac{1}{4}} \cdot 5^{\frac{1}{2}} = 10^{\frac{1}{4}} \cdot (4 \cdot 10)^{\frac{1}{4}} \cdot 5^{\frac{1}{2}} = 10^{\frac{1}{4}} \cdot 4^{\frac{1}{4}} \cdot 10^{\frac{1}{4}} \cdot 5^{\frac{1}{2}} = 10^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}} = 10^{\frac{1}{2}} \cdot 10^{\frac{1}{2}} = 10$ 3) $(0,0016)^{-\frac{3}{4}} - (0,04)^{-\frac{1}{2}} + (0,216)^{-\frac{2}{3}} = (\frac{16}{10000})^{-\frac{3}{4}} - (\frac{4}{100)^{-\frac{1}{2}}} + (\frac{216}{1000})^{-\frac{2}{3}} = (\frac{2}{10})^{\frac{3}{4} \cdot 4} - (\frac{2}{10})^{\frac{1}{2} \cdot 2} + (\frac{6}{10})^{\frac{2}{3} \cdot 3} = (\frac{2}{10})^3 - (\frac{2}{10})^1 + (\frac{6}{10})^2 = (\frac{10}{2})^3 - (\frac{10}{2})^1 + (\frac{10}{6})^2 = 5^3 - 5 + (\frac{5}{3})^2 = 125 - 5 + \frac{25}{9} = 120 + \frac{25}{9} = \frac{1080+25}{9} = \frac{1105}{9}$ 4) $\frac{32^{0,24} \cdot 4^{0,7}}{64^{0,6} \cdot 16^{0,25}} = \frac{(2^5)^{0,24} \cdot (2^2)^{0,7}}{(2^6)^{0,6} \cdot (2^4)^{0,25}} = \frac{2^{1,2} \cdot 2^{1,4}}{2^{3,6} \cdot 2^1} = \frac{2^{2,6}}{2^{4,6}} = 2^{-2} = \frac{1}{4}$ 5) $\frac{12^{\frac{1}{2}}}{7^{\frac{2}{3}} \cdot 8^{\frac{1}{6}}} : \frac{32^{\frac{1}{5}} \cdot 7^{\frac{5}{3}}}{8^{\frac{1}{2}}} = \frac{(3 \cdot 4)^{\frac{1}{2}}}{7^{\frac{2}{3}} \cdot 2^{\frac{1}{2}}} : \frac{2 \cdot 7^{\frac{5}{3}}}{2^{\frac{3}{2}}} = \frac{3^{\frac{1}{2}} \cdot 2}{7^{\frac{2}{3}} \cdot 2^{\frac{1}{2}}} \cdot \frac{2^{\frac{3}{2}}}{2 \cdot 7^{\frac{5}{3}}} = \frac{3^{\frac{1}{2}} \cdot 2^{\frac{3}{2}}}{7^{\frac{2}{3}} \cdot 2^{\frac{1}{2}} \cdot 2 \cdot 7^{\frac{5}{3}}} = \frac{3^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}}{7^{\frac{7}{3}} \cdot 2} = \frac{3^{\frac{1}{2}}}{7^{\frac{7}{3}} \cdot 2^{\frac{1}{2}}}$ 6) $(\frac{5^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}}{15^{\frac{2}{3}} \cdot 2^{\frac{16}{3}}})^{-1,5} = (\frac{5^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}}{(3 \cdot 5)^{\frac{2}{3}} \cdot 2^{\frac{16}{3}}})^{-1,5} = (\frac{5^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}} \cdot 5^{\frac{2}{3}} \cdot 2^{\frac{16}{3}}})^{-1,5} = (2^{-\frac{16}{3}})^{-1,5} = 2^{\frac{16}{3} \cdot \frac{3}{2}} = 2^8 = 256$ Надеюсь, я помог тебе разобраться!