### Instruction: Find the product of $1011_2 \cdot 101_2$.  Express your answer in base 2.### Response: I know that multiplying by a power of 2 is equivalent to shifting the digits to the left, so I can use the distributive property to split up the multiplication into two parts: $1011_2 \cdot 101_2 = 1011_2 \cdot (100_2 + 1_2) = 1011_2 \cdot 100_2 + 1011_2 \cdot 1_2$.
To multiply $1011_2$ by $100_2$, I just need to shift the digits of $1011_2$ two places to the left, which gives me $101100_2$.
To multiply $1011_2$ by $1_2$, I just need to copy the digits of $1011_2$, which gives me $1011_2$.
To add $101100_2$ and $1011_2$, I need to align the digits and perform binary addition, which involves carrying over when the sum of two digits is 2 or more.
I start from the rightmost digits and work my way to the left:
\begin{align*}
 & \quad \quad \quad 101100_2 \\
+& \quad \quad \quad 001011_2 \\
=& \quad \quad \quad \underline{110111_2}
\end{align*}
The final answer is $110111_2$.