To simplify the binary expression \( 101011_2 + 10111_2 - 10101_2 \), we will first convert each binary number to decimal, perform the calculations in decimal, and then convert the result back to binary if necessary.
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Convert each binary number to decimal:
\[ 101011_2 = 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 32 + 0 + 8 + 0 + 2 + 1 = 43_{10} \]
\[ 10111_2 = 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 16 + 0 + 4 + 2 + 1 = 23_{10} \]
\[ 10101_2 = 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 16 + 0 + 4 + 0 + 1 = 21_{10} \]
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Perform the operations in decimal: \[ 43 + 23 - 21 = 65 \]
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Convert the result back to binary: To convert \( 65_{10} \) to binary, we divide by 2 and keep track of the remainders.
- \( 65 \div 2 = 32 \) remainder 1
- \( 32 \div 2 = 16 \) remainder 0
- \( 16 \div 2 = 8 \) remainder 0
- \( 8 \div 2 = 4 \) remainder 0
- \( 4 \div 2 = 2 \) remainder 0
- \( 2 \div 2 = 1 \) remainder 0
- \( 1 \div 2 = 0 \) remainder 1
Reading the remainders from bottom to top gives us \( 1000001_2 \).
Thus, the final simplified result is:
\[ 1000001_2 \]