What is the quotient \( \frac{-8 a^{8} b^{-2}}{10 a^{-4} b^{-10}} \) in simplified form? Assume \( a \neq 0, b \neq 0 \)

To simplify the expression \( \frac{-8 a^{8} b^{-2}}{10 a^{-4} b^{-10}} \), we can start by dividing the coefficients and then using the properties of exponents. First, we divide the coefficients: \[ \frac{-8}{10} = -\frac{4}{5} \] Next, we handle the exponents for \( a \): \[ \frac{a^{8}}{a^{-4}} = a^{8 - (-4)} = a^{8 + 4} = a^{12} \] Now, for the \( b \) terms: \[ \frac{b^{-2}}{b^{-10}} = b^{-2 - (-10)} = b^{-2 + 10} = b^{8} \] Combining these results, we have: \[ \frac{-8 a^{8} b^{-2}}{10 a^{-4} b^{-10}} = -\frac{4}{5} a^{12} b^{8} \] So the simplified form of the quotient is: \[ -\frac{4}{5} a^{12} b^{8} \]