To solve the given expression:

$\left(a^3b^4\right)^{-2}\left(a^{-3}b^{-5}\right)^{-4}$

We need to simplify step by step:

Step 1: Simplify each part of the expression

1. First term: $\left(a^3b^4\right)^{-2} = a^{3 \cdot (-2)}b^{4 \cdot (-2)} = a^{-6}b^{-8}$

2. Second term: $\left(a^{-3}b^{-5}\right)^{-4} = a^{-3 \cdot (-4)}b^{-5 \cdot (-4)} = a^{12}b^{20}$

Step 2: Combine the two results

Now combine the two simplified terms: $a^{-6}b^{-8} \cdot a^{12}b^{20}$

Using the laws of exponents ($a^m \cdot a^n = a^{m+n}$ and $b^m \cdot b^n = b^{m+n}$): $a^{-6 + 12}b^{-8 + 20} = a^6b^{12}$

Final Answer:

The simplified expression is: $a^6b^{12}$

This corresponds to option (b).

Would you like additional details or further explanation on this problem?

Related Questions:

1. How do you simplify expressions with negative exponents?
2. What is the general rule for multiplying terms with the same base?
3. Why does $\left(a^m\right)^n = a^{m \cdot n}$?
4. How do exponents distribute over products like $(ab)^n$?
5. Can you provide a real-world example involving exponents?

Tip:

Always simplify each term separately before combining them in exponent problems to reduce errors.