55.

\[ \frac{17 x^{-8} \cdot 28 y}{14 y^{-12} \cdot 51 x^{-21}} = \frac{17 \cdot 28}{14 \cdot 51} \cdot \frac{x^{-8}}{x^{-21}} \cdot \frac{y}{y^{-12}} \]
\[ \frac{17 \cdot (2 \cdot 14)}{14 \cdot (3 \cdot 17)} = \frac{2}{3} \]
\[ x^{-8 - (-21)} = x^{-8+21} = x^{13} \]
\[ y^{1 - (-12)} = y^{1+12} = y^{13} \]
\[ \text{Результат: } \frac{2}{3} x^{13} y^{13} \]
\[ -1.6 m^{-4} n^3 \cdot (-2 m^{-3} p^{-6})^{-3} = -1.6 m^{-4} n^3 \cdot (-2)^{-3} (m^{-3})^{-3} (p^{-6})^{-3} \]
\[ = -1.6 m^{-4} n^3 \cdot (-\frac{1}{8}) m^9 p^{18} \]
\[ = (-1.6 \cdot -\frac{1}{8}) \cdot (m^{-4} \cdot m^9) \cdot n^3 \cdot p^{18} \]
\[ = 0.2 m^{-4+9} n^3 p^{18} = 0.2 m^5 n^3 p^{18} = \frac{1}{5} m^5 n^3 p^{18} \]
\[ 2\frac{1}{4} a^{-5} b \cdot \left( 1\frac{1}{2} a^{-1} b^{-3} \right)^{-3} = \frac{9}{4} a^{-5} b \cdot \left( \frac{3}{2} a^{-1} b^{-3} \right)^{-3} \]
\[ = \frac{9}{4} a^{-5} b \cdot \left( \frac{3}{2} \right)^{-3} (a^{-1})^{-3} (b^{-3})^{-3} = \frac{9}{4} a^{-5} b \cdot \frac{8}{27} a^3 b^9 \]
\[ = \left( \frac{9}{4} \cdot \frac{8}{27} \right) \cdot (a^{-5} \cdot a^3) \cdot (b \cdot b^9) = \frac{9 \cdot 8}{4 \cdot 27} a^{-5+3} b^{1+9} = \frac{72}{108} a^{-2} b^{10} = \frac{2}{3} a^{-2} b^{10} = \frac{2 b^{10}}{3 a^2} \]
\[ (-10 a^{-2} b c^{-11})^{-2} \cdot (0.1 b c^{-2})^{-3} = (-10)^{-2} (a^{-2})^{-2} b^{-2} (c^{-11})^{-2} \cdot (0.1)^{-3} b^{-3} (c^{-2})^{-3} \]
\[ = \frac{1}{100} a^4 b^{-2} c^{22} \cdot \frac{1}{(0.1)^3} b^{-3} c^6 = \frac{1}{100} a^4 b^{-2} c^{22} \cdot \frac{1}{0.001} b^{-3} c^6 = \frac{1}{100} a^4 b^{-2} c^{22} \cdot 1000 b^{-3} c^6 \]
\[ = (\frac{1}{100} \cdot 1000) \cdot a^4 \cdot (b^{-2} \cdot b^{-3}) \cdot (c^{22} \cdot c^6) = 10 a^4 b^{-2+(-3)} c^{22+6} = 10 a^4 b^{-5} c^{28} = \frac{10 a^4 c^{28}}{b^5} \]
\[ \left( -\frac{1}{5} a^{-3} b^{-7} \right)^{-3} \cdot (-5 a^2 b^6)^{-2} = (-1)^{-3} (\frac{1}{5})^{-3} (a^{-3})^{-3} (b^{-7})^{-3} \cdot (-5)^{-2} (a^2)^{-2} (b^6)^{-2} \]
\[ = -1 \cdot 5^3 a^9 b^{21} \cdot \frac{1}{25} a^{-4} b^{-12} = -125 a^9 b^{21} \cdot \frac{1}{25} a^{-4} b^{-12} \]
\[ = (-125 \cdot \frac{1}{25}) \cdot (a^9 \cdot a^{-4}) \cdot (b^{21} \cdot b^{-12}) = -5 a^{9+(-4)} b^{21+(-12)} = -5 a^5 b^9 \]
\[ \left( \frac{8 p^{-4}}{q^{-1}} \right)^{-2} \cdot (16 p^{-6} q^3)^3 = (8 p^{-4})^{-2} (q^{-1})^{-2} \cdot 16^3 (p^{-6})^3 (q^3)^3 \]
\[ = 8^{-2} (p^{-4})^{-2} q^2 \cdot 4096 p^{-18} q^9 = \frac{1}{64} p^8 q^2 \cdot 4096 p^{-18} q^9 \]
\[ = (\frac{1}{64} \cdot 4096) \cdot (p^8 \cdot p^{-18}) \cdot (q^2 \cdot q^9) = 64 p^{8+(-18)} q^{2+9} = 64 p^{-10} q^{11} = \frac{64 q^{11}}{p^{10}} \]

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