If x = (1111)₂, y = (1001)₂ and z = (110)₂, then what is x³ - y³ - z³ - 3xyz equal to?
(0)₂
We are given that \( x = (1111)_2 \), \( y = (1001)_2 \), and \( z = (110)_2 \), and we are asked to find the value of \( x^3 - y^3 - z^3 - 3xyz \). First, we convert the binary numbers to decimal form: - \( x = (1111)_2 = 1 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 8 + 4 + 2 + 1 = 15 \), - \( y = (1001)_2 = 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 8 + 1 = 9 \), - \( z = (110)_2 = 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0 = 4 + 2 = 6 \). Now, we substitute the decimal values of \( x \), \( y \), and \( z \) into the expression \( x^3 - y^3 - z^3 - 3xyz \): \[ x^3 - y^3 - z^3 - 3xyz = 15^3 - 9^3 - 6^3 - 3 \cdot 15 \cdot 9 \cdot 6. \] First, calculate the cubes: \[ 15^3 = 3375, \quad 9^3 = 729, \quad 6^3 = 216. \] Next, calculate the product \( 3 \cdot 15 \cdot 9 \cdot 6 \): \[ 3 \cdot 15 \cdot 9 \cdot 6 = 2430. \] Now, substitute these values into the expression: \[ 3375 - 729 - 216 - 2430 = 3375 - 729 - 216 - 2430 = 3375 - 945 - 2430 = 3375 - 3375 = 0. \] Therefore, the value of \( x^3 - y^3 - z^3 - 3xyz \) is \( 0 \).
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