First you could call this "distributing" the exponent, not sure what the real name is for this step.
\(\frac{(-2x^{-2})^3(6x)^2}{2(-3x^{-1})^3}=\frac{(-2^3x^{-2*3})(6^2x^2)}{2(-3^3x^{-1*3})}=\frac{(-8x^{-6})(36x^2)}{2(-27x^{-3})}\)
Now all terms with negative exponents in the numerator go to the denominator and lose the negative sign and vice versa.
\(\frac{(-8x^3)(36x^2)}{2x^6(-27)}\)
Now multiply the things in the numerator and the things in the denominator together.
\(\frac{-288x^{3+2}}{-54x^6}=\frac{-288x^{5}}{-54x^6}\)
Now reduce the fraction by 18x5
\(\frac{-288x^{5}}{-54x^6}=\frac{16}{3x^{6-5}}=\frac{16}{3x}\)
Same as guest. 
