The expression you need to simplify is:

$\left( 7a^9b^{-5} \right)^2$

Step-by-Step Solution:

1. Apply the exponent to each factor: When raising a product to an exponent, we distribute the exponent to each factor within the parentheses. This gives us:

   $7^2 \cdot (a^9)^2 \cdot (b^{-5})^2$

2. Simplify each part:

   • $7^2 = 49$
   • $(a^9)^2 = a^{9 \times 2} = a^{18}$
   • $(b^{-5})^2 = b^{-5 \times 2} = b^{-10}$

3. Combine the results: $49a^{18}b^{-10}$

4. Eliminate negative exponents: To get rid of the negative exponent for $b$, we move it to the denominator: $\frac{49a^{18}}{b^{10}}$

Final Simplified Expression:

$\frac{49a^{18}}{b^{10}}$

If you need further clarification or have any questions, feel free to ask! Here are some related questions:

1. What is the law of exponents used to simplify expressions like this one?
2. How do negative exponents relate to fractions?
3. Can the exponent rule also be applied to fractional exponents?
4. What happens when you raise a product to a negative exponent?
5. How do you handle variables with exponents when they appear in a fraction?

Tip: Always remember that negative exponents mean "take the reciprocal" of the base.