To evaluate [tex]\(\left(111_{\text{two}}\right)^2 - \left(101_{\text{two}}\right)^2\)[/tex], we will first convert the binary numbers to their decimal (base 10) equivalents, then calculate the squares of each number, and finally find the difference between these squares.
1. Convert the binary numbers to decimal:
- The binary number [tex]\(111_2\)[/tex] can be converted to decimal as follows:
[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]
- The binary number [tex]\(101_2\)[/tex] can be converted to decimal as follows:
[tex]\[ 101_2 = 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 \][/tex]
[tex]\[ = 1 \cdot 4 + 0 \cdot 2 + 1 \cdot 1 \][/tex]
[tex]\[ = 4 + 0 + 1 \][/tex]
[tex]\[ = 5 \][/tex]
2. Calculate the square of each decimal number:
- For [tex]\(7\)[/tex]:
[tex]\[ 7^2 = 49 \][/tex]
- For [tex]\(5\)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
3. Find the difference between the squares:
[tex]\[ 49 - 25 = 24 \][/tex]
Therefore, the value of [tex]\(\left(111_{\text{two}}\right)^2 - \left(101_{\text{two}}\right)^2\)[/tex] is [tex]\(\boxed{24}\)[/tex].
