Simplify the expression $(-3ab)^2 \times 4a \div a^3b$.

1. Problem Description

Simplify the expression $(-3ab)^2 \times 4a \div a^3b$.

2. Solution Steps

First, we simplify the term $(-3ab)^2$. Recall the power of a product rule: $(xy)^n = x^n y^n$.

$(-3ab)^2 = (-3)^2 \times a^2 \times b^2 = 9a^2b^2$

Now, we substitute this result into the original expression:

$9a^2b^2 \times 4a \div a^3b$

The expression becomes:

$9a^2b^2 \times 4a \div a^3b = \frac{9a^2b^2 \times 4a}{a^3b}$

Multiply the numerator:

$9a^2b^2 \times 4a = 36a^3b^2$

So, the expression now looks like:

$\frac{36a^3b^2}{a^3b}$

We can now simplify the expression by dividing:

$\frac{36a^3b^2}{a^3b} = 36 \times \frac{a^3}{a^3} \times \frac{b^2}{b}$

Recall the quotient of powers rule: $\frac{x^m}{x^n} = x^{m-n}$.

$\frac{a^3}{a^3} = a^{3-3} = a^0 = 1$

$\frac{b^2}{b} = b^{2-1} = b^1 = b$

Thus, we have:

$36 \times 1 \times b = 36b$

3. Final Answer

$36b$