### 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 in base 2 is equivalent to shifting the digits to the left by that power.
For example, $101_2 \cdot 10_2 = 1010_2$, which is shifting the digits one place to the left.
So I can use the distributive property to write $1011_2 \cdot 101_2 = 1011_2 \cdot (100_2 + 1_2) = 1011_2 \cdot 100_2 + 1011_2 \cdot 1_2$.
To multiply $1011_2$ by $100_2$, I just shift the digits two places to the left: $1011_2 \cdot 100_2 = 101100_2$.
To multiply $1011_2$ by $1_2$, I just leave the digits unchanged: $1011_2 \cdot 1_2 = 1011_2$.
To add these two products, I align the digits and add them column by column, starting from the right.
I carry over any excess bits to the next column.
For example, $1_2 + 1_2 = 10_2$, so I write $0_2$ in the sum and carry over $1_2$ to the next column.
So I get $101100_2 + 1011_2 = 110111_2$.