Evaluate $(101_{two})^3$

Analyze the question: We need to cube the binary number 101 and give the answer in binary.

Concept: Convert to base 10, cube it, then convert back to base 2. (Or multiply directly in binary.)

Solution:

Step 1: Convert 101₂ to base 10: 1×2² + 0×2¹ + 1×2⁰ = 4 + 0 + 1 = 5

Step 2: Cube it: 5³ = 125

Step 3: Convert 125 to binary by repeated division by 2:

• 125 ÷ 2 = 62 remainder 1
• 62 ÷ 2 = 31 remainder 0
• 31 ÷ 2 = 15 remainder 1
• 15 ÷ 2 = 7 remainder 1
• 7 ÷ 2 = 3 remainder 1
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top: 1111101₂

Check: 64 + 32 + 16 + 8 + 4 + 0 + 1 = 125 ✓

Compare with the options:

• A. 111101₂ = 61 ✗
• B. 11111101₂ = 253 ✗
• C. 1111101₂ = 125 ✓
• D. 11001₂ = 25 ✗

Tip: In binary, 101₂ is 5, so for powers, convert to base 10 first. Memorise the powers of 2 (1, 2, 4, 8, 16, 32, 64, 128) so you can quickly check conversions: 125 = 128 − 3, which is 1111101 in binary.