278. Упростите выражение:

Решение:

1. \( 3a^{-3} \cdot 4a^{4} = (3 \cdot 4) \cdot (a^{-3} \cdot a^{4}) = 12 \cdot a^{-3+4} = 12a^{1} = 12a \)
2. \( \frac{10b^{-4}}{15b^{-5}} = \frac{10}{15} \cdot \frac{b^{-4}}{b^{-5}} = \frac{2}{3} \cdot b^{-4 - (-5)} = \frac{2}{3} \cdot b^{-4+5} = \frac{2}{3} b^{1} = \frac{2b}{3} \)
3. \( (2c^{-6})^{4} = 2^{4} \cdot (c^{-6})^{4} = 16 \cdot c^{-6 \cdot 4} = 16c^{-24} \)
4. \( m^{2}n \cdot mn^{-2} = (m^{2} \cdot m) \cdot (n \cdot n^{-2}) = m^{2+1} \cdot n^{1-2} = m^{3} n^{-1} = \frac{m^{3}}{n} \)
5. \( abc^{-1} \cdot ab^{-1}c = (a \cdot a) \cdot (b \cdot b^{-1}) \cdot (c^{-1} \cdot c) = a^{2} \cdot b^{1-1} \cdot c^{-1+1} = a^{2} b^{0} c^{0} = a^{2} \)
6. \( \frac{kp^{-6}}{k^{4}p^{4}} = \frac{k}{k^{4}} \cdot \frac{p^{-6}}{p^{4}} = k^{1-4} \cdot p^{-6-4} = k^{-3} p^{-10} = \frac{1}{k^{3}p^{10}} \)
7. \( (c^{-6}d^{2})^{-7} = (c^{-6})^{-7} \cdot (d^{2})^{-7} = c^{-6 \cdot (-7)} d^{2 \cdot (-7)} = c^{42} d^{-14} = \frac{c^{42}}{d^{14}} \)
8. \( \frac{1}{3}a^{-3}b^{-6} \cdot \frac{6}{7}a^{7}b^{4} = \left(\frac{1}{3} \cdot \frac{6}{7}\right) \cdot (a^{-3} \cdot a^{7}) \cdot (b^{-6} \cdot b^{4}) = \frac{6}{21} \cdot a^{-3+7} \cdot b^{-6+4} = \frac{2}{7} a^{4} b^{-2} = \frac{2a^{4}}{7b^{2}} \)
9. \( 0.2c^{-3}d^{5} \cdot 1.5c^{-2}d^{-5} = (0.2 \cdot 1.5) \cdot (c^{-3} \cdot c^{-2}) \cdot (d^{5} \cdot d^{-5}) = 0.3 \cdot c^{-3-2} \cdot d^{5-5} = 0.3 c^{-5} d^{0} = \frac{0.3}{c^{5}} \)
10. \( 4x^{8} \cdot (-3x^{-2}y^{4})^{-2} = 4x^{8} \cdot ((-3)^{-2} (x^{-2})^{-2} (y^{4})^{-2}) = 4x^{8} \cdot (\frac{1}{9} x^{4} y^{-8}) = (4 \cdot \frac{1}{9}) \cdot (x^{8} \cdot x^{4}) \cdot y^{-8} = \frac{4}{9} x^{12} y^{-8} = \frac{4x^{12}}{9y^{8}} \)
11. \( \frac{13m^{-10}}{12n^{-8}} \cdot \frac{27n}{26m^{2}} = \frac{13 \cdot 27}{12 \cdot 26} \cdot \frac{m^{-10}}{m^{2}} \cdot \frac{n^{-8}}{n^{-1}} = \frac{1 \cdot 9}{4 \cdot 2} \cdot m^{-10-2} \cdot n^{-8-(-1)} = \frac{9}{8} m^{-12} n^{-7} = \frac{9}{8m^{12}n^{7}} \)
12. \( \frac{18p^{-6}k^{2}}{7} : \frac{15k^{-2}}{p^{6}} = \frac{18p^{-6}k^{2}}{7} \cdot \frac{p^{6}}{15k^{-2}} = \frac{18}{7 \cdot 15} \cdot \frac{p^{-6}}{p^{6}} \cdot \frac{k^{2}}{k^{-2}} = \frac{6}{35} \cdot p^{-6-6} \cdot k^{2-(-2)} = \frac{6}{35} p^{-12} k^{4} = \frac{6k^{4}}{35p^{12}} \)