The expression in the image is:

$\left( \frac{23a^3b^8}{4a^9b^4} \right)^{-2}.$

Step-by-Step Simplification:

1. Simplify the fraction inside the parentheses: $\frac{23a^3b^8}{4a^9b^4}.$

   • Simplify the $a$-terms: $\frac{a^3}{a^9} = a^{3-9} = a^{-6}.$
   • Simplify the $b$-terms: $\frac{b^8}{b^4} = b^{8-4} = b^4.$
   • Combine the constants: $\frac{23}{4}.$ The fraction becomes: $\frac{23b^4}{4a^6}.$

2. Apply the exponent $-2$: The entire fraction is raised to the power of $-2$: $\left( \frac{23b^4}{4a^6} \right)^{-2}.$ Recall the rule for negative exponents: $\left( \frac{x}{y} \right)^{-n} = \left( \frac{y}{x} \right)^n$. Therefore: $\left( \frac{23b^4}{4a^6} \right)^{-2} = \left( \frac{4a^6}{23b^4} \right)^2.$

3. Simplify by squaring the numerator and denominator:

   • Square the numerator: $(4a^6)^2 = 4^2 \cdot a^{6 \cdot 2} = 16a^{12}.$
   • Square the denominator: $(23b^4)^2 = 23^2 \cdot b^{4 \cdot 2} = 529b^8.$ The result becomes: $\frac{16a^{12}}{529b^8}.$

Final Answer:

$\boxed{\frac{16a^{12}}{529b^8}}$

Do you have any questions or want further details?

Here are 5 related questions to explore further:

1. What is the rule for dividing exponents with the same base?
2. How do negative exponents affect expressions?
3. What happens when a fraction is raised to a negative power?
4. Can you rewrite the final answer using a single negative exponent?
5. How would the answer change if the initial exponent was $-3$ instead of $-2$?

Tip: Always simplify the inner fraction or expression before applying any negative or fractional exponent rules to avoid errors.