Simplify the expression. Assume that all bases are not equal to 0 . \( \left(a^{-5} b^{4}\right)^{-7} \)

To simplify the expression \( \left(a^{-5} b^{4}\right)^{-7} \), we can apply the power of a product rule, which states that \( (xy)^n = x^n y^n \). This means we can distribute the exponent of \(-7\) to both \(a^{-5}\) and \(b^{4}\): \[ \left(a^{-5} b^{4}\right)^{-7} = (a^{-5})^{-7} (b^{4})^{-7} \] Now, simplifying each part: \[ (a^{-5})^{-7} = a^{35} \] \[ (b^{4})^{-7} = b^{-28} \] Putting it all together, we have: \[ a^{35} b^{-28} \] This can also be written as: \[ \frac{a^{35}}{b^{28}} \] So, the simplified form of the expression is \( \frac{a^{35}}{b^{28}} \).