If x = (1111)₂ , y = (1001)₂ and z = (110)₂ , then what is x^3 - y^3 - z^3 - 3xyz equal to?

Step-by-step solution

Converting the given binary numbers to decimal: x = (1111)₂ = 1 2^3 + 1 2^2 + 1 2^1 + 1 2^0 = 15 y = (1001)₂ = 1 2^3 + 0 2^2 + 0 2^1 + 1 2^0 = 9 z = (110)₂ = 1 2^2 + 1 2^1 + 0 2^0 = 6 Observing the relationship between x , y , and z : y + z = 9 + 6 = 15 = x x - y - z = 0 Using the algebraic identity, if a + b + c = 0 , then a^3 + b^3 + c^3 - 3abc = 0 . Substituting a = x , b = -y , and c = -z : x + (-y) + (-z) = 0 x^3 + (-y)^3 + (-z)^3 - 3(x)(-y)(-z) = 0 x^3 - y^3 - z^3 - 3xyz = 0 In binary representation, 0 is (0)