Quick questions on Roots of complex equations explained: H2 Mathematics Vectors and Complex Numbers

The $n$ distinct $n$th roots of $w = r e^{i\theta}$ are

An equation such as $z^n = w$ is solved directly by the roots formula. More general polynomials are solved by factoring (using known or conjugate roots) into linear and quadratic factors, then solving each. The fundamental theorem guarantees a degree-$n$ polynomial has exactly $n$ roots counted with multiplicity.

The roots are spaced exactly $\frac{2\pi}{n}$ apart; an uneven set signals an arithmetic slip.

How many distinct fifth roots does a nonzero complex number have, and how are they arranged? [2 marks]

Find the square roots of $i$ in exponential form. [3 marks]

State the modulus of each cube root of $27$. [1 mark]