Which expression is equivalent to \( \frac{(5 a b)^{3}}{30 a^{-6} b^{-7}} \) ? Assume \( a \neq 0, b \neq 0 \)

First, let's simplify the expression \( \frac{(5 a b)^{3}}{30 a^{-6} b^{-7}} \). Start by expanding the numerator: \[ (5 a b)^{3} = 5^{3} a^{3} b^{3} = 125 a^{3} b^{3} \] Next, rewrite the denominator: \[ 30 a^{-6} b^{-7} = 30 \cdot \frac{1}{a^{6}} \cdot \frac{1}{b^{7}} = \frac{30}{a^{6} b^{7}} \] Now, rewrite the original expression: \[ \frac{125 a^{3} b^{3}}{30 a^{-6} b^{-7}} = 125 a^{3} b^{3} \cdot \frac{a^{6} b^{7}}{30} \] By applying the properties of exponents: \[ = \frac{125 a^{3 + 6} b^{3 + 7}}{30} = \frac{125 a^{9} b^{10}}{30} \] Now, simplify the fraction \( \frac{125}{30} \): \[ \frac{125}{30} = \frac{25}{6} \] Putting it all together, we get: \[ \frac{125 a^{9} b^{10}}{30} = \frac{25 a^{9} b^{10}}{6} \] Thus, the expression \( \frac{(5 a b)^{3}}{30 a^{-6} b^{-7}} \) simplifies to: \[ \frac{25 a^{9} b^{10}}{6} \]