We need to simplify the expression:

$\frac{a^2 \times (b^2)^6}{a^4 \times b^9}$

Step-by-Step Solution:

1. Simplify the powers of $b$:

   • $(b^2)^6 = b^{2 \times 6} = b^{12}$.

   Now, the expression becomes:

   $\frac{a^2 \times b^{12}}{a^4 \times b^9}$

2. Simplify the powers of $a$:

   • Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$, we have:

   $\frac{a^2}{a^4} = a^{2 - 4} = a^{-2}$

3. Simplify the powers of $b$:

   • Again, using the property of exponents $\frac{b^m}{b^n} = b^{m-n}$, we have:

   $\frac{b^{12}}{b^9} = b^{12 - 9} = b^3$

4. Combine the simplified terms:

   • The expression now becomes:

   $a^{-2} \times b^3$

5. Final Simplified Expression:

   • The negative exponent $a^{-2}$ can be rewritten as $\frac{1}{a^2}$, so the expression can also be written as:

   $\frac{b^3}{a^2}$

Final Answer:

$\frac{b^3}{a^2}$

Would you like further explanation on any step, or do you have any questions?

Here are 5 related questions you might find useful:

1. How do you simplify expressions with multiple exponents?
2. What are the rules for multiplying and dividing powers of the same base?
3. How do you handle negative exponents in algebraic expressions?
4. What happens when you raise a power to another power in an expression?
5. How can you simplify a more complex algebraic fraction?

Tip: Always apply exponent rules step by step when simplifying, ensuring you handle each variable separately to avoid mistakes.