We will simplify the expression $(4a^3b^{-3})^2$ using the index laws.

Step-by-step:

1. Distribute the exponent:
   The exponent of 2 is applied to every factor inside the parentheses. This means we need to square each term: $(4a^3b^{-3})^2 = 4^2 \cdot (a^3)^2 \cdot (b^{-3})^2$

2. Simplify the individual terms:

   • $4^2 = 16$
   • $(a^3)^2 = a^{3 \times 2} = a^6$
   • $(b^{-3})^2 = b^{-3 \times 2} = b^{-6}$

Thus, the expression simplifies to: $16a^6b^{-6}$

Final Answer:

$16a^6b^{-6}$

Would you like to dive deeper into how the index laws work or do you have any other questions?

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Here are some related questions you might find interesting:

1. How do you simplify an expression with multiple terms using index laws?
2. What is the rule for multiplying terms with the same base and exponents?
3. How do you handle negative exponents in index notation?
4. Can you explain the process of expanding binomials with exponents?
5. What are the laws of indices when dividing terms with the same base?

Tip: When applying the exponent to a product, remember to distribute it to each factor inside the parentheses.