How many times does the variable $a$ appear in the product $(3a)(4b^2)(2ab)$? What single power of $a$ represents that many factors of $a$ multiplied together?

You have $b^2$ in one factor and a single $b$ in another. When you multiply $b^2$ by $b$, how do the exponents combine? Recall $x^m \cdot x^n = x^{m+n}$.

Pick simple values for the variables, such as $a=2$ and $b=3$. In Desmos, type the original expression with these values substituted: (3*2)*(4*3^2)*(2*2*3) and note the numeric result.

For each answer choice, substitute $a=2$ and $b=3$ and enter the resulting expression into Desmos (for example, for choice A: 24*2*3^3). Compare each result to the value from the original expression; the choice whose value matches is the equivalent expression.

Rewrite the product so you can clearly see the numerical coefficients and the variables:

$(3a)(4b^2)(2ab)$ has

• numerical parts: $3$, $4$, and $2$
• variable parts: $a$, $b^2$, and $ab$.

Multiply the numerical parts first:

$3 \cdot 4 \cdot 2 = 24$

So the coefficient of the simplified expression will be $24$.

Count how many times $a$ appears in the product.

You have one $a$ in $(3a)$ and one $a$ in $(2ab)$, so you are multiplying $a \cdot a$.

Using exponents, $a \cdot a = a^2$.

Now combine the $b$ factors.

You have $b^2$ in $(4b^2)$ and one $b$ in $(2ab)$, so you are multiplying $b^2 \cdot b$.

Using exponents, $b^2 \cdot b = b^{2+1} = b^3$.

Putting everything together, the simplified expression is $24a^2 b^3$, which matches answer choice C.