Find z²⁶ in Cartesian form.

01 / Standard#Your turn

z=\frac{\sqrt3+i}{1+i}

Official paper · jm02-2024 · 4(a)(ii) · PDF 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Raise the modulus to 26 and multiply the argument by 26.

Hint 2

Reduce −13π/6 modulo 2π.

Worked solution

Apply De Moivre’s theorem.
$z^{26}=2^{13}\operatorname{cis}(-13\pi/6)=2^{13}\operatorname{cis}(-\pi/6)$
Evaluate the exact trigonometric values.
$z^{26}=2^{12}\sqrt3-2^{12}i$

4096√3−4096i.

Checks and common pitfalls: The modulus is (√2)²⁶=2¹³, not 2²⁶.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Apply De Moivre’s theorem.
$z^{26}=2^{13}\operatorname{cis}(-13\pi/6)=2^{13}\operatorname{cis}(-\pi/6)$
Evaluate the exact trigonometric values.
$z^{26}=2^{12}\sqrt3-2^{12}i$

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

2024 JM02