We are tasked with simplifying the expression using the product rule for exponents. The given expression is:

$\left(\frac{a^9 b^4}{a^6 b}\right)$

Step-by-Step Solution:

We will apply the product rule for exponents, which states that when multiplying powers with the same base, you add the exponents.

$\text{Product Rule: } a^m \times a^n = a^{m+n}$

1. Simplifying for $a$:

$\frac{a^9}{a^6} = a^{9-6} = a^3$

2. Simplifying for $b$:

$\frac{b^4}{b^1} = b^{4-1} = b^3$

Now that we've simplified both parts:

$a^3 b^3$

Final Answer:

The simplified expression is:

$a^3 b^3$

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

Here are 5 related questions:

1. What is the product rule for exponents when multiplying powers with the same base?
2. How would the answer change if the exponents of $a$ were reversed?
3. What happens when you divide powers with the same base?
4. How can you simplify expressions involving negative exponents?
5. How can you apply the product rule when variables have different bases?

Tip: Always check if you can simplify terms with the same base using exponent rules before calculating the values directly.