A Pure Imaginary Bottom

When the bottom is only imaginary, like 2i, its real part is 0, so its conjugate is -2i. That works: (2i)(-2i) = -4i^2, which is 4, a real number.

Multiplying by i works too, with smaller numbers: (2i)(i) = 2i^2, which is -2. Any nonzero real multiple of i makes the bottom real.

Take (8 - 6i)/(2i). Times i on top and bottom: the top is 8i - 6i^2, which is 6 + 8i, and the bottom is -2. Split the fraction: 6 over -2 is -3, and 8i over -2 is -4i, so the answer is -3 - 4i. The bottom is negative, so every sign of the top flips.