Answer
The common ratio of the $\{a_nb_n\}$ sequence is the product of the common ratios of the two given geometric sequences.
Work Step by Step
We are given the geometric sequences:
$$\begin{align*}
\{a_n\}&\text{ with common ratio }r\\
\{b_n\}&\text{ with common ratio }q.\end{align*}$$
We have to study the sequence:
$$c_n=a_nb_n.$$
We calculate the ratio between two consecutive terms:
$$\begin{align*}
\dfrac{c_{k+1}}{c_k}&=\dfrac{a_{k+1}b_{k+1}}{a_kb_k}\\
&=\dfrac{a_{k+1}}{a_k}\cdot \dfrac{b_{k+1}}{b_k}\\
&=rq.
\end{align*}$$
We got the $c_{k+1}/c_k=rq$, therefore constant. It follows that the sequence $\{c_n\}$ is also a geometric sequence with common ratio equal to the product of the common ratios of the two initial geometric sequences.