Compute the 2024th power for each of the two nonreal roots from part (a).

01 / Standard#Your turn

z=-1\pm\sqrt3i

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use z³=8 and the remainder of 2024 modulo 3.

Hint 2

2024=3·674+2.

Worked solution

Reduce the large exponent to a square.
$z^{2024}=8^{674}z^2=2^{2022}z^2$
Square each conjugate and preserve the linked sign.
$(-1\pm\sqrt3i)^2=-2\mp2\sqrt3i\implies z^{2024}=-2^{2023}(1\pm\sqrt3i)$

For z=−1±√3i, z²⁰²⁴=−2²⁰²³(1±√3i), using matching signs.

Checks and common pitfalls: The outer negative sign changes both real and imaginary components.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Reduce the large exponent to a square.
$z^{2024}=8^{674}z^2=2^{2022}z^2$
Square each conjugate and preserve the linked sign.
$(-1\pm\sqrt3i)^2=-2\mp2\sqrt3i\implies z^{2024}=-2^{2023}(1\pm\sqrt3i)$

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

2025 JM02