Answer :
Certainly! Let's evaluate the given expression [tex]\(\left(1101_{\text {two }}\right)^2 - \left(111_{\text {two }}\right)^2\)[/tex] step-by-step.
### Step 1: Convert Binary to Decimal
1. Convert [tex]\(1101_2\)[/tex] to decimal:
- [tex]\(1101_2 = 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\)[/tex]
- [tex]\( = 1 \cdot 8 + 1 \cdot 4 + 0 \cdot 2 + 1 \cdot 1\)[/tex]
- [tex]\( = 8 + 4 + 0 + 1\)[/tex]
- [tex]\( = 13\)[/tex]
Thus, [tex]\(1101_2\)[/tex] is equivalent to [tex]\(13_{10}\)[/tex].
2. Convert [tex]\(111_2\)[/tex] to decimal:
- [tex]\(111_2 = 1 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0\)[/tex]
- [tex]\( = 1 \cdot 4 + 1 \cdot 2 + 1 \cdot 1\)[/tex]
- [tex]\( = 4 + 2 + 1\)[/tex]
- [tex]\( = 7\)[/tex]
Thus, [tex]\(111_2\)[/tex] is equivalent to [tex]\(7_{10}\)[/tex].
### Step 2: Square the Decimal Numbers
1. Square [tex]\(13\)[/tex]:
- [tex]\(13^2 = 169\)[/tex]
2. Square [tex]\(7\)[/tex]:
- [tex]\(7^2 = 49\)[/tex]
### Step 3: Perform the Subtraction
1. Subtract the squared values:
- [tex]\(169 - 49 = 120\)[/tex]
### Step 4: Convert the Result Back to Binary
1. Convert [tex]\(120_{10}\)[/tex] to binary:
- Divide [tex]\(120\)[/tex] by [tex]\(2\)[/tex], keeping track of the remainders:
- [tex]\(120 \div 2 = 60\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(60 \div 2 = 30\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(30 \div 2 = 15\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(15 \div 2 = 7\)[/tex] remainder [tex]\(1\)[/tex]
- [tex]\(7 \div 2 = 3\)[/tex] remainder [tex]\(1\)[/tex]
- [tex]\(3 \div 2 = 1\)[/tex] remainder [tex]\(1\)[/tex]
- [tex]\(1 \div 2 = 0\)[/tex] remainder [tex]\(1\)[/tex]
- Reading the remainders in reverse order gives [tex]\(1111000_2\)[/tex].
### Conclusion
Thus, [tex]\(\left(1101_{\text{two}}\right)^2 - \left(111_{\text{two}}\right)^2 = 1111000_{\text{two}}\)[/tex].
The correct answer is:
[tex]\[ \boxed{1111000_{two}} \][/tex]
### Step 1: Convert Binary to Decimal
1. Convert [tex]\(1101_2\)[/tex] to decimal:
- [tex]\(1101_2 = 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\)[/tex]
- [tex]\( = 1 \cdot 8 + 1 \cdot 4 + 0 \cdot 2 + 1 \cdot 1\)[/tex]
- [tex]\( = 8 + 4 + 0 + 1\)[/tex]
- [tex]\( = 13\)[/tex]
Thus, [tex]\(1101_2\)[/tex] is equivalent to [tex]\(13_{10}\)[/tex].
2. Convert [tex]\(111_2\)[/tex] to decimal:
- [tex]\(111_2 = 1 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0\)[/tex]
- [tex]\( = 1 \cdot 4 + 1 \cdot 2 + 1 \cdot 1\)[/tex]
- [tex]\( = 4 + 2 + 1\)[/tex]
- [tex]\( = 7\)[/tex]
Thus, [tex]\(111_2\)[/tex] is equivalent to [tex]\(7_{10}\)[/tex].
### Step 2: Square the Decimal Numbers
1. Square [tex]\(13\)[/tex]:
- [tex]\(13^2 = 169\)[/tex]
2. Square [tex]\(7\)[/tex]:
- [tex]\(7^2 = 49\)[/tex]
### Step 3: Perform the Subtraction
1. Subtract the squared values:
- [tex]\(169 - 49 = 120\)[/tex]
### Step 4: Convert the Result Back to Binary
1. Convert [tex]\(120_{10}\)[/tex] to binary:
- Divide [tex]\(120\)[/tex] by [tex]\(2\)[/tex], keeping track of the remainders:
- [tex]\(120 \div 2 = 60\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(60 \div 2 = 30\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(30 \div 2 = 15\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(15 \div 2 = 7\)[/tex] remainder [tex]\(1\)[/tex]
- [tex]\(7 \div 2 = 3\)[/tex] remainder [tex]\(1\)[/tex]
- [tex]\(3 \div 2 = 1\)[/tex] remainder [tex]\(1\)[/tex]
- [tex]\(1 \div 2 = 0\)[/tex] remainder [tex]\(1\)[/tex]
- Reading the remainders in reverse order gives [tex]\(1111000_2\)[/tex].
### Conclusion
Thus, [tex]\(\left(1101_{\text{two}}\right)^2 - \left(111_{\text{two}}\right)^2 = 1111000_{\text{two}}\)[/tex].
The correct answer is:
[tex]\[ \boxed{1111000_{two}} \][/tex]
