The expression provided is:

$\left( \frac{3a^2}{b^3} \right)^2$

We are tasked with simplifying this.

Step-by-step solution:

1. Apply the power rule: $\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$. This gives: $\frac{(3a^2)^2}{(b^3)^2}$

2. Simplify the numerator and denominator:

   • $(3a^2)^2 = 3^2 \cdot (a^2)^2 = 9a^4$
   • $(b^3)^2 = b^{6}$

Thus, the expression simplifies to:

$\frac{9a^4}{b^6}$

Final Answer:

The correct simplified expression is:

$\boxed{\frac{9a^4}{b^6}}$

Looking at the options provided in the image, this matches the third option.

Would you like more details, or do you have any questions?

Here are five related questions to deepen your understanding:

1. What is the general rule for simplifying powers of fractions?
2. How does the exponent rule $(a^m)^n = a^{mn}$ apply to both numerators and denominators?
3. How would the simplification change if the exponent were negative?
4. Can you apply this technique to an expression like $\left(\frac{5x^3}{y^2}\right)^3$?
5. How does this method differ when variables are in both the numerator and denominator?

Tip: When simplifying expressions with exponents, apply the exponent to both the numerator and denominator separately before combining terms.