Powers and roots | Year 12

01 / LEARN

The idea, step by step

One equation, several roots

For $z^{n} = R cis θ$ with $R > 0$ and positive integer $n$ , the modulus of each root is $R^{1 / n}$ . Its angle must satisfy $n ϕ = θ + 2 k π$ . Thus the roots are $R^{1 / n} cis ((θ + 2 k π) / n)$ for $k = 0, \dots, n - 1$ .

Exactly $n$ distinct roots arise: successive arguments differ by $2 π / n$ , making a regular polygon on a circle centred at the origin. Increasing $k$ by $n$ merely adds a full turn to a root. If the right side is zero, the sole distinct root is zero.

z_{k} = R^{1 / n} cis (\frac{θ + 2 k π}{n}), k = 0, \dots, n - 1

Do not divide only the principal angle

Dividing a single principal argument gives only one of the roots. The extra whole turns must be introduced before division. Afterwards, angles may be reduced to a requested interval.

A root is checked by raising its modulus to $n$ and multiplying its angle by $n$ . Listing the expected count, with distinct angles modulo $2 π$ , helps prevent omission or duplication.

A common trap

Writing only $R^{1 / n} cis (θ / n)$ as all the roots.

Instead: Add $2 k π$ before dividing the angle by n, then select n distinct values.

02 / WORKED EXAMPLES

See the thinking

Example 1

Cube roots of a negative real number

Solve $z^{3} = - 8$ .

Write $- 8 = 8 cis π$ .
Each modulus is 2 and the arguments are $(π + 2 k π) / 3$ for $k = 0, 1, 2$ .
These three angles are distinct modulo a full turn.

z = 2 cis (π / 3), 2 cis π, 2 cis (5 π / 3)

, or

1 + \sqrt{3} i, - 2, 1 - \sqrt{3} i

Example 2

Four roots of a non-real number

Solve $z^{4} = 16 i$ in polar form.

$16 i = 16 cis (π / 2)$ .
The fourth-root modulus is 2; divide $π / 2 + 2 k π$ by 4.
The angle step is $π / 2$ , producing four distinct roots.

z = 2 cis (π / 8 + k π / 2)

k = 0, 1, 2, 3

Check your understanding

How many distinct roots does $z^{5} = 32$ have, and what is their common modulus?

Show answer

Five roots, each of modulus 2.

The nonzero right side gives five distinct equally spaced arguments; $32^{1 / 5} = 2$ .

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Solve $z^{2} = - 9$ .
2 marks
List the cube roots of 8 in polar form.
2 marks
Find all solutions of $z^{4} = - 16$ .
3 marks
Write the roots of $z^{3} = 8 i$ in principal-argument polar form.
3 marks
Explain why $k = 4$ gives no new root in $z_{k} = 2 cis (π / 8 + k π / 2)$ .
2 marks
The three cube roots of a nonzero number form an equilateral triangle. Explain using arguments.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Solve $z^{2} = - 9$ .
2 marks
Use arguments $(π + 2 k π) / 2$ for $k = 0, 1$ and modulus 3.
$z = 3 i$ or $z = - 3 i$ .
List the cube roots of 8 in polar form.
2 marks
The angles are $2 k π / 3$ for $k = 0, 1, 2$ .
$2, 2 cis (2 π / 3), 2 cis (4 π / 3)$ .
Find all solutions of $z^{4} = - 16$ .
3 marks
The modulus is 2. Arguments come from $(π + 2 k π) / 4$ .
$2 cis (π / 4 + k π / 2)$ , $k = 0, 1, 2, 3$ .
Write the roots of $z^{3} = 8 i$ in principal-argument polar form.
3 marks
Angles from $π / 6 + 2 k π / 3$ are $π / 6, 5 π / 6, 3 π / 2$ ; reduce the last by $2 π$ .
$2 cis (π / 6), 2 cis (5 π / 6), 2 cis (- π / 2)$ .
Explain why $k = 4$ gives no new root in $z_{k} = 2 cis (π / 8 + k π / 2)$ .
2 marks
The angle for $k = 4$ is $π / 8 + 2 π$ , the same direction as $π / 8$ .
It repeats the $k = 0$ root.
The three cube roots of a nonzero number form an equilateral triangle. Explain using arguments.
3 marks
Equal central angles on one circle give equal chord lengths, so all three sides are equal.
Adjacent roots differ in angle by $2 π / 3$ and have equal modulus.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics complex numbers: Cartesian and polar representations, de Moivre, roots and roots of unity, Argand geometry and loci, complex factorisation and conjugate roots of real polynomials. Real-polynomial techniques are taught in the separate Real Polynomials chapter.

Official curriculum source ↗

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