1. Stating the problem: Simplify the expression $$[(101001)_2 + (1111)_2] - (100110)_2$$ where the numbers are in binary. 2. Convert each binary number to decimal: $$ (101001)_2 = 1\times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 32 + 0 + 8 + 0 + 0 + 1 = 41 $$ $$ (1111)_2 = 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 4 + 2 + 1 = 15 $$ $$ (100110)_2 = 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 32 + 0 + 0 + 4 + 2 + 0 = 38 $$ 3. Perform the operations in decimal: $$ (101001)_2 + (1111)_2 = 41 + 15 = 56 $$ $$ 56 - (100110)_2 = 56 - 38 = 18 $$ 4. Convert the final result back to binary: $$ 18_{10} = (10010)_2 $$ Answer: The simplified binary result is $$ (10010)_2 $$.