Complex Numbers

Powers of $i$ repeat every 4. Since $27 = 24 + 3$ and $i^{24} = 1$, the answer is $i^3 = -i$.

Add the real parts and the imaginary parts separately.

$8 - 12i + 2i - 3i^2 = 8 - 10i + 3 = 11 - 10i$. The $-3i^2$ becomes $+3$.

$\sqrt{25 + 144} = \sqrt{169} = 13$. The modulus is the distance from the origin on an Argand diagram.

Multiply top and bottom by $3-i$: $\dfrac{2(3-i)}{9+1} = \dfrac{6-2i}{10} = \dfrac35 - \dfrac15 i$.

The conjugate $1+2i$ is also a root, so $(x-1+2i)(x-1-2i) = (x-1)^2 + 4 = x^2 - 2x + 5$.