Step 1: Write each number as the product of prime factors. \( \begin{aligned} 27 & =\square \cdot \square \cdot \square \\ 45 & =3 \cdot 3 \cdot \square\end{aligned} \)

Let's break down those numbers into their prime factors! For 27, we can express it as \(3 \cdot 3 \cdot 3\) or \(3^3\). So, the completed equation would be: \(27 = 3 \cdot 3 \cdot 3\). Now for 45, we already have two of the numbers as \(3 \cdot 3\) which is \(3^2\). The remaining factor can be found by dividing 45 by 9 (which is \(3 \cdot 3\)): \(45 = 3 \cdot 3 \cdot 5\). So the completed equation is: \(45 = 3 \cdot 3 \cdot 5\). Putting it all together: \[ \begin{aligned} 27 & = 3 \cdot 3 \cdot 3 \\ 45 & = 3 \cdot 3 \cdot 5 \end{aligned} \]