Finding (1 + i)⁸

Trigonometry · Axiom Academy

EXAMPLE Finding (1 + i)⁸ using DeMoivre's Theorem Learn to convert complex numbers to polar form and apply DeMoivre's theorem to find powers efficiently. Excellent work! You've successfully used DeMoivre's Theorem to find (1 + i)^8. Here's what we learned: Polar Form is Powerful: Converting to polar form makes raising to powers much simpler than expanding (1 + i)^8 using binomial theorem! DeMoivre's Theorem: For any complex number r·cis(θ), we have (r·cis(θ))^n = r^n·cis(nθ). Raise the modulus to the power and multiply the angle by the power. Angle Simplification: Always reduce angles to their simplest form. Here, 2π = 0, showing that multiplying the angle by 8 brought us full circle back to the positive real axis. Pattern Recognition: Powers of 1 + i follow a pattern: i^4 = 1, so (1 + i)^8 lands on the real axis. The angle π/4 multiplied by 8 gives exactly 2π, completing one full rotation. Real Result from Complex Base: Even though we started with a complex number, raising it to the 8th power gave us a purely real result (16). This happens when the angle becomes a multiple of 2π. DeMoivre's Theorem is essential for finding powers and roots of complex numbers efficiently. Look for opportunities to use it whenever you need to raise a complex number to an integer power!

This is the written version of the interactive lesson above. See the full Trigonometry course.