The expression given is:

$\left(4a^7b^{-3}\right)^{-2}$

Step 1: Apply the power to each term inside the parentheses.

The expression can be rewritten by distributing the exponent -2 to each term inside the parentheses:

$(4)^{-2} \cdot \left(a^7\right)^{-2} \cdot \left(b^{-3}\right)^{-2}$

Step 2: Simplify each term.

• $(4)^{-2} = \frac{1}{4^2} = \frac{1}{16}$
• $\left(a^7\right)^{-2} = a^{7 \times (-2)} = a^{-14} = \frac{1}{a^{14}}$
• $\left(b^{-3}\right)^{-2} = b^{-3 \times (-2)} = b^{6}$

Step 3: Combine the terms.

Now, combine all the terms together:

$\frac{1}{16} \cdot \frac{1}{a^{14}} \cdot b^6 = \frac{b^6}{16a^{14}}$

Final Answer:

$\boxed{\frac{b^6}{16a^{14}}}$

Would you like any additional details or have any questions?

Here are some related questions you might find helpful:

1. How do you simplify expressions with negative exponents?
2. What are the rules for distributing exponents over products?
3. How do you convert negative exponents into positive ones?
4. Can you provide another example of simplifying an expression with exponents?
5. What happens if the base of the exponent is a fraction?

Tip: When dealing with negative exponents, remember that they indicate reciprocation of the base.