Simplify: (11$_{two}$)$^2$

Analyze the question: We need to square the binary number 11₂ and give the answer in base two.

Concept: Convert to base ten, square it, then convert back to binary. You can also multiply directly in binary.

Solution:

Step 1: Convert 11₂ to base ten. 11₂ = (1 × 2¹) + (1 × 2⁰) = 2 + 1 = 3

Step 2: Square it. 3² = 9

Step 3: Convert 9 to base two.

• 9 ÷ 2 = 4 remainder 1
• 4 ÷ 2 = 2 remainder 0
• 2 ÷ 2 = 1 remainder 0
• 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top: 9 = 1001₂

Check by direct binary multiplication:

    11
  × 11
  ----
    11
   110
  ----
  1001

This gives 1001₂, which matches.

Answer: Option A, 1001₂.

Tip: For binary questions, convert to base ten, do the arithmetic, and convert back. Then check with the options. Also remember that 11₂ × 11₂ is like (2+1)² = 4 + 2·2 + 1 = 9, which is 8 + 1 = 1001₂.