54. Упростите выражение: 1) (1/3) * p^-2 * q^-5 * (5/9) * p^6 * q^3; 2) -0.4*b^-3*c^7 * 1.5*b^2*c^-6; 3) 0.45*m^-3*n^2*p^-4 * 1(1/9)*m^8*n^-11*p^6; 4) 5*a^-6 * (-3*a^-2*b^3)^-2.

54.

• \[ \frac{1}{3} p^{-2} q^{-5} \cdot \frac{5}{9} p^6 q^3 = \left( \frac{1}{3} \cdot \frac{5}{9} \right) \cdot (p^{-2} \cdot p^6) \cdot (q^{-5} \cdot q^3) = \frac{5}{27} p^{-2+6} q^{-5+3} = \frac{5}{27} p^4 q^{-2} = \frac{5 p^4}{27 q^2} \]
• \[ -0.4 b^{-3} c^7 \cdot 1.5 b^2 c^{-6} = (-0.4 \cdot 1.5) \cdot (b^{-3} \cdot b^2) \cdot (c^7 \cdot c^{-6}) = -0.6 b^{-3+2} c^{7+(-6)} = -0.6 b^{-1} c^1 = -\frac{0.6 c}{b} \]
• \[ 0.45 m^{-3} n^2 p^{-4} \cdot \frac{1}{9} m^8 n^{-11} p^6 = (0.45 \cdot \frac{1}{9}) \cdot (m^{-3} \cdot m^8) \cdot (n^2 \cdot n^{-11}) \cdot (p^{-4} \cdot p^6) = \frac{0.45}{9} m^{-3+8} n^{2+(-11)} p^{-4+6} = 0.05 m^5 n^{-9} p^2 = \frac{0.05 m^5 p^2}{n^9} \]
• \[ 5 a^{-6} \cdot (-3 a^{-2} b^3)^{-2} = 5 a^{-6} \cdot (-3)^{-2} (a^{-2})^{-2} (b^3)^{-2} = 5 a^{-6} \cdot \frac{1}{9} a^{4} b^{-6} = \left( 5 \cdot \frac{1}{9} \right) \cdot (a^{-6} \cdot a^4) \cdot b^{-6} = \frac{5}{9} a^{-6+4} b^{-6} = \frac{5}{9} a^{-2} b^{-6} = \frac{5}{9 a^2 b^6} \]