Express the following fraction in simplest form, only using positive exponents. \[ \frac{\left(-3 q^{-3}\right)^{3}}{-6 q^{-6}} \]

To simplify the expression \(\frac{\left(-3 q^{-3}\right)^{3}}{-6 q^{-6}}\), let's break it down step by step. First, simplify the numerator: \[ \left(-3 q^{-3}\right)^{3} = (-3)^{3} \cdot (q^{-3})^{3} = -27 \cdot q^{-9} \] Now substituting this back into the fraction gives us: \[ \frac{-27 q^{-9}}{-6 q^{-6}} \] Since both the numerator and the denominator have a negative sign, they cancel out: \[ \frac{27 q^{-9}}{6 q^{-6}} \] Now, simplify the constants and the \(q\) terms. Simplifying the constants: \[ \frac{27}{6} = \frac{9}{2} \] Next, for the \(q\) terms, we have: \[ \frac{q^{-9}}{q^{-6}} = q^{-9 - (-6)} = q^{-9 + 6} = q^{-3} \] Putting this together, we obtain: \[ \frac{9}{2} q^{-3} \] To write \(q^{-3}\) with a positive exponent, we convert it: \[ q^{-3} = \frac{1}{q^{3}} \] Therefore: \[ \frac{9}{2} q^{-3} = \frac{9}{2 q^{3}} \] So, the final simplified form of the original expression is: \[ \frac{9}{2 q^{3}} \]