2022HSCShort answer

Q13 (c) (i)

2 marks

Question

Consider the equation $z^5+1=0$ , where $z$ is a complex number. Solve the equation $z^5+1=0$ by finding the fifth roots of $-1$ .

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Writes $-1$ in general exponential form, or equivalent merit	1

Sample answer:
$z^5+1=0$
$z^5=-1=e^{i\pi}$
$z=e^{i(2k\pi+\pi)/5},\quad k=0,\pm1,\pm2$
$z=e^{i\pi/5},\ e^{i3\pi/5},\ e^{-i\pi/5},\ e^{i\pi/5},\ z=e^{i\pi}=-1$
$z=-1,\ e^{\pm i\pi/5},\ e^{\pm i3\pi/5}$

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Syllabus alignment

Complex Numbers

Solving equations with complex numbersUse De Moivre's theorem and solve equations and polynomial problems involving complex and conjugate roots.

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