Fully simplify the expression below. \[ h^{9} k^{-4} \times 4 h^{6} k^{-5} \div\left(10 h^{2} k^{-3}\right)^{-2} \] Give your answer as a fraction without any negative indices.

To simplify the expression \[ h^{9} k^{-4} \times 4 h^{6} k^{-5} \div\left(10 h^{2} k^{-3}\right)^{-2}, \] let's break it down step-by-step. 1. **Simplifying the divisor**: First, calculate \((10 h^{2} k^{-3})^{-2}\): \[ (10 h^{2} k^{-3})^{-2} = 10^{-2} (h^{2})^{-2} (k^{-3})^{-2} = \frac{1}{100} h^{-4} k^{6}. \] This is equivalent to: \[ \frac{h^{-4} k^{6}}{100}. \] 2. **Rewriting the division**: Now, rewrite the entire expression: \[ h^{9} k^{-4} \times 4 h^{6} k^{-5} \div \frac{h^{-4} k^{6}}{100} = h^{9} k^{-4} \times 4 h^{6} k^{-5} \times \frac{100}{h^{-4} k^{6}}. \] 3. **Multiplying by the reciprocal**: Combine the terms: \[ h^{9} k^{-4} \times 4 h^{6} k^{-5} \times 100 = 400 h^{9} h^{6} k^{-4} k^{-5}. \] 4. **Combine like bases**: Combine \(h\) and \(k\): \[ 400 h^{9+6} k^{-4-5} = 400 h^{15} k^{-9}. \] 5. **Getting rid of the negative exponent**: To eliminate the negative exponent, we write: \[ 400 h^{15} k^{-9} = \frac{400 h^{15}}{k^{9}}. \] Thus, the fully simplified expression is: \[ \frac{400 h^{15}}{k^{9}}. \]