278. Упростите выражение: 1) 3a⁻³ ⋅ 4a⁴; 2) 10b⁻⁴ / 15b⁻⁵; 3) (2c⁻⁶)⁴; 4) m⁻²n ⋅ mn⁻²; 5) abc⁻¹ ⋅ ab⁻¹c; 6) kp⁻⁶ / k⁴p⁴; 7) (c⁻⁶d²)⁻⁷; 8) ⅓ a⁻³b⁻⁶ ⋅ ⁶⁄₇ a⁷b⁴; 9) 0,2c⁻³d⁵ ⋅ 1,5c⁻²d⁻⁵; 10) 4x⁸ ⋅ (-3x⁻²y⁴)⁻²; 11) (13m⁻¹⁰ / 12n⁻⁸) ⋅ (27n / 26m²); 12) (18p⁻⁶k² / 7) : (15k⁻² / p⁶)

Решение:

Используем свойства степеней:

• \( a^m \cdot a^n = a^{m+n} \)
• \( a^m : a^n = a^{m-n} \)
• \( (a^m)^n = a^{m \cdot n} \)
• \( (ab)^n = a^n b^n \)
• \( (a/b)^n = a^n / b^n \)
• \( a^{-n} = 1/a^n \)

1. \( 3a^{-3} \cdot 4a^4 = (3 \cdot 4) \cdot (a^{-3} \cdot a^4) = 12 \cdot a^{-3+4} = 12a^1 = 12a \)
2. \( \frac{10b^{-4}}{15b^{-5}} = \frac{10}{15} \cdot \frac{b^{-4}}{b^{-5}} = \frac{2}{3} \cdot b^{-4-(-5)} = \frac{2}{3} b^{-4+5} = \frac{2}{3} b^1 = \frac{2b}{3} \)
3. \( (2c^{-6})^4 = 2^4 \cdot (c^{-6})^4 = 16 \cdot c^{-6 \cdot 4} = 16c^{-24} \)
4. \( m^{-2}n \cdot mn^{-2} = (m^{-2} \cdot m^1) \cdot (n^1 \cdot n^{-2}) = m^{-2+1} \cdot n^{1+(-2)} = m^{-1}n^{-1} = \frac{1}{mn} \)
5. \( abc^{-1} \cdot ab^{-1}c = (a \cdot a) \cdot (b \cdot b^{-1}) \cdot (c^{-1} \cdot c^1) = a^2 \cdot b^{1+(-1)} \cdot c^{-1+1} = a^2 b^0 c^0 = a^2 \)
6. \( \frac{kp^{-6}}{k^4p^4} = k^{1-4} \cdot p^{-6-4} = k^{-3}p^{-10} = \frac{1}{k^3p^{10}} \)
7. \( (c^{-6}d^2)^{-7} = (c^{-6})^{-7} \cdot (d^2)^{-7} = c^{42}d^{-14} = \frac{c^{42}}{d^{14}} \)
8. \( \frac{1}{3}a^{-3}b^{-6} \cdot \frac{6}{7}a^7b^4 = (\frac{1}{3} \cdot \frac{6}{7}) \cdot (a^{-3} \cdot a^7) \cdot (b^{-6} \cdot b^4) = \frac{6}{21} \cdot a^{-3+7} \cdot b^{-6+4} = \frac{2}{7} a^4 b^{-2} = \frac{2a^4}{7b^2} \)
9. \( 0.2c^{-3}d^5 \cdot 1.5c^{-2}d^{-5} = (0.2 \cdot 1.5) \cdot (c^{-3} \cdot c^{-2}) \cdot (d^5 \cdot d^{-5}) = 0.3 \cdot c^{-3+(-2)} \cdot d^{5+(-5)} = 0.3 c^{-5} d^0 = 0.3c^{-5} = \frac{0.3}{c^5} \)
10. \( 4x^8 \cdot (-3x^{-2}y^4)^{-2} = 4x^8 \cdot (-3)^{-2} \cdot (x^{-2})^{-2} \cdot (y^4)^{-2} = 4x^8 \cdot \frac{1}{9} \cdot x^4 \cdot y^{-8} = \frac{4}{9} \cdot x^{8+4} \cdot y^{-8} = \frac{4}{9} x^{12} y^{-8} = \frac{4x^{12}}{9y^8} \)
11. \( \frac{13m^{-10}}{12n^{-8}} \cdot \frac{27n}{26m^2} = \frac{13 \cdot 27}{12 \cdot 26} \cdot \frac{m^{-10} \cdot n}{n^{-8} \cdot m^2} = \frac{13 \cdot 3 \cdot 9}{12 \cdot 2 \cdot 13} \cdot m^{-10-2} \cdot n^{1-(-8)} = \frac{3 \cdot 9}{12 \cdot 2} \cdot m^{-12} n^9 = \frac{27}{24} m^{-12} n^9 = \frac{9}{8} m^{-12} n^9 = \frac{9n^9}{8m^{12}} \)
12. \( \frac{18p^{-6}k^2}{7} : \frac{15k^{-2}}{p^6} = \frac{18p^{-6}k^2}{7} \cdot \frac{p^6}{15k^{-2}} = \frac{18 \cdot 1}{7 \cdot 15} \cdot \frac{p^{-6} \cdot p^6 \cdot k^2}{k^{-2}} = \frac{18}{105} \cdot p^{-6+6} \cdot k^{2-(-2)} = \frac{6}{35} \cdot p^0 \cdot k^4 = \frac{6k^4}{35} \)

Ответ: 1) \( 12a \); 2) \( \frac{2b}{3} \); 3) \( 16c^{-24} \); 4) \( \frac{1}{mn} \); 5) \( a^2 \); 6) \( \frac{1}{k^3p^{10}} \); 7) \( \frac{c^{42}}{d^{14}} \); 8) \( \frac{2a^4}{7b^2} \); 9) \( \frac{0.3}{c^5} \); 10) \( \frac{4x^{12}}{9y^8} \); 11) \( \frac{9n^9}{8m^{12}} \); 12) \( \frac{6k^4}{35} \).