Find the nonreal cube roots of eight.

01 / Standard#Your turn

z=a+bi,\quad a,b\in\mathbb R,\ b\ne0,\quad z^3=8

Official paper · jm02-2025 · 4(a) · PDF 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Factor z³−8.

Hint 2

Discard the real root before solving the quadratic factor.

Worked solution

z=−1±√3i.

Checks and common pitfalls: The root 2 must be excluded because the imaginary part is nonzero.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Use the difference-of-cubes identity.
$z^3-8=(z-2)(z^2+2z+4)$
The root 2 violates b≠0; solve the other factor.
$z=-1\pm\sqrt3i$

Think first. Reveal a hint when the class is ready.

2025 JM02