The expression given in the image is:

$\left( \frac{3a^5b^5}{ab^2} \right)^2$

We need to simplify this expression. Let's break it down step by step:

Step 1: Simplify inside the parentheses

We can simplify the fraction inside the parentheses: $\frac{3a^5b^5}{ab^2} = 3 \cdot \frac{a^5}{a} \cdot \frac{b^5}{b^2}$

• For the $a$-terms: $\frac{a^5}{a} = a^{5-1} = a^4$
• For the $b$-terms: $\frac{b^5}{b^2} = b^{5-2} = b^3$

So, the simplified expression inside the parentheses becomes: $3a^4b^3$

Step 2: Apply the exponent outside the parentheses

Now we need to square this entire expression: $\left( 3a^4b^3 \right)^2$

This means we need to square each term inside: $3^2 = 9,\quad (a^4)^2 = a^{4 \cdot 2} = a^8,\quad (b^3)^2 = b^{3 \cdot 2} = b^6$

Thus, the fully simplified expression is: $9a^8b^6$

Final Answer:

The correct choice is A: $9a^8b^6$.

Would you like more details on this? Feel free to ask any questions! Here are some related questions for further exploration:

1. How do you simplify expressions with variables raised to powers?
2. What is the general rule for simplifying powers of products?
3. How would the process change if there were negative exponents involved?
4. How does squaring affect coefficients and exponents differently?
5. What would the result be if only the $b$-term had an exponent outside?

Tip: When simplifying expressions involving exponents, always remember to apply exponent rules such as $(a^m / a^n) = a^{m-n}$ and $(a^m)^n = a^{m \cdot n}$.