### 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 bits to the left by that amount.
For example, multiplying by $10_2$ is the same as shifting one bit to the left, and multiplying by $100_2$ is the same as shifting two bits to the left.
So, I can use the distributive property to break up the product as follows: $1011_2 \cdot 101_2 = 1011_2 \cdot (100_2 + 1_2) = 1011_2 \cdot 100_2 + 1011_2 \cdot 1_2$.
Now, I can shift the bits to get $1011_2 \cdot 100_2 = 101100_2$ and $1011_2 \cdot 1_2 = 1011_2$.
To add these two terms, I need to align them by their rightmost bits and then add them column by column, carrying over any extra bits as needed.
This gives me:

   101100
  +  1011
  ------
  110111