If x = (1111)_2 , y = (1001)_2 and z = (110)_2 , then what is x^3 -...

Converting the given binary numbers to decimal:

$x = (1111)_2 = 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 15$

$y = (1001)_2 = 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 9$

$z = (110)_2 = 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 6$

Observing the relationship between $x$, $y$, and $z$:

$y + z = 9 + 6 = 15 = x$

$\Rightarrow x - y - z = 0$

Using the algebraic identity, if $a + b + c = 0$, then $a^3 + b^3 + c^3 - 3abc = 0$.

Substituting $a = x$, $b = -y$, and $c = -z$:

$x + (-y) + (-z) = 0$

$\Rightarrow x^3 + (-y)^3 + (-z)^3 - 3(x)(-y)(-z) = 0$

$\Rightarrow x^3 - y^3 - z^3 - 3xyz = 0$

In binary representation, $0$ is $(0)_2$.

Answer: $(0)_2$