Roots of complex numbers

Finding nth roots

To solve zⁿ = w, write w = Re^iφ. Because adding 2π to the argument gives the same number, w = Re^(i(φ + 2kπ)) for any integer k. Then

z = R^1/n e^(i(φ + 2kπ)/n), for k = 0, 1, ..., n - 1.

All the roots have modulus R^1/n, and their arguments are spaced 2π/n apart.

Roots of unity

The solutions of zⁿ = 1 are 1, ω, ω², ..., ω^n-1, where ω = e^2πi/n.

They lie on the unit circle at the vertices of a regular n-sided polygon, and their sum is zero: 1 + ω + ... + ω^n-1 = 0.

Exam technique

• Take the nth root of the modulus; the June 2025 examiners' report notes a few students raising it to the power n instead.
• Give arguments in the range asked for, usually -π < θ ≤ π, adjusting by 2π.
• Find all n roots, not just one.
• A quick Argand sketch shows the polygon and catches mistakes.

Worked example

Solve z³ = 8i, giving answers in the form re^iθ with -π < θ ≤ π.

1. 8i = 8e^iπ/2 = 8e^(i(π/2 + 2kπ)).
2. z = 2e^(i(π/2 + 2kπ)/3) = 2e^(i(π/6 + 2kπ/3)).
3. k = 0: π/6; k = 1: 5π/6; k = 2: 3π/2, which becomes -π/2.

Answer: z = 2e^iπ/6, 2e^5iπ/6, 2e^-iπ/2

Check yourself

1. Find the modulus of each root of z⁵ = 16 + 16√3 i.
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   2. |16 + 16√3 i| = √(256 + 768) = 32. Each root has modulus 32^1/5 = 2.

2. The root of z⁴ = -16 with the smallest positive argument has argument kπ. Find k. Give your answer as a fraction in its simplest form.
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   1/4. -16 = 16e^iπ. The arguments are (π + 2kπ)/4: π/4, 3π/4, -3π/4, -π/4. The smallest positive one is π/4.

3. The roots of z⁵ = 16 + 16√3 i are written 2e^iθ with -π < θ ≤ π. The most negative θ is kπ. Find k. Give your answer as a fraction in its simplest form.
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   -11/15. arg w = π/3, so θ = (π/3 + 2kπ)/5 = π/15, 7π/15, 13π/15, 19π/15, 25π/15. Subtracting 2π from the last two gives -11π/15 and -π/3. The most negative is -11π/15.

4. Find the sum of the six sixth roots of unity.
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   0. The roots are symmetric about the origin (a regular hexagon), so they add to 0. Algebraically: 1 + ω + ... + ω⁵ = (ω⁶ - 1)/(ω - 1) = 0.

5. The roots of z⁴ = 81 are joined to form a polygon in the Argand diagram. Find its area.
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   18. The roots are 3, 3i, -3 and -3i: a square with diagonals of length 6. Area = ½ × 6 × 6 = 18.

6. ω is a complex cube root of unity (ω ≠ 1). What is 1 + ω + ω²?
   • 1
   • -1
   • 0
   • 3
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   0. ω³ = 1, so ω³ - 1 = (ω - 1)(ω² + ω + 1) = 0, and ω ≠ 1.

7. Find the real part of the root of z³ = 8i that lies in the first quadrant, to 3 decimal places.
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   1.732. The root is 2e^iπ/6 = 2(cos(π/6) + i sin(π/6)) = √3 + i. Real part √3 = 1.732.

8. How many distinct roots does z⁷ = 1 + i have?
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   7. An equation zⁿ = w (w ≠ 0) has exactly n distinct roots.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit F2
• Pearson Edexcel IAL Further Pure Mathematics 2 scheme of work
• Pearson Edexcel IAL Further Pure Mathematics F2 (WFM02) examiners' report, June 2025

Last updated 8 November 2026. Spotted a mistake? Tell us.