Let's simplify the expression step by step:

Expression:

$\left(-\frac{1}{2} a^{-3} b^9\right)\left(8a^{-1} b^{-4}\right)$

Step 1: Multiply the constants

1. Multiply the numerical coefficients: $-\frac{1}{2} \times 8 = -4$

Step 2: Multiply the $a$ terms

1. Use the property of exponents $a^m \times a^n = a^{m+n}$: $a^{-3} \times a^{-1} = a^{-3-1} = a^{-4}$

Step 3: Multiply the $b$ terms

1. Again, use the property of exponents: $b^9 \times b^{-4} = b^{9-4} = b^5$

Step 4: Combine everything

Now, combine the results from steps 1, 2, and 3: $-4 \times a^{-4} \times b^5$

Final Answer:

$-4a^{-4}b^5$

Or, if you want to express $a^{-4}$ with a positive exponent: $-\frac{4b^5}{a^4}$

Would you like more details, or do you have any questions?

---

Related Questions:

1. How would the expression change if the $a$ terms had positive exponents initially?
2. What if one of the exponents was zero—how would that affect the result?
3. Can you simplify an expression with more complex exponents, such as fractional exponents?
4. How do you simplify expressions with more than two variables?
5. What are the general rules for multiplying powers of the same base?

Tip: When multiplying expressions with exponents, always add the exponents if the bases are the same, but be careful with the signs of the exponents.