(1+i)5

Jasonbradley

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Jun 16, 2024

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DeMoivre's Theorem and (1 + i)^5

The expression (1 + i)^5 can be solved using DeMoivre's Theorem, which is a powerful tool for calculating powers of complex numbers.

Understanding DeMoivre's Theorem

DeMoivre's Theorem states that for any complex number z = r(cos θ + i sin θ) and any integer n, the following holds:

z^n = r^n(cos nθ + i sin nθ)

Essentially, this means to raise a complex number to a power, we raise the modulus (r) to that power and multiply the angle (θ) by the power.

Solving (1 + i)^5

1. Convert to Polar Form:

   • Find the modulus: r = √(1² + 1²) = √2
   • Find the angle: θ = arctan(1/1) = π/4

   Therefore, 1 + i = √2(cos(π/4) + i sin(π/4))

2. Apply DeMoivre's Theorem:

   (1 + i)^5 = [√2(cos(π/4) + i sin(π/4))]^5 (1 + i)^5 = (√2)^5(cos(5π/4) + i sin(5π/4))

3. Simplify:

   (1 + i)^5 = 4√2(-√2/2 - i√2/2) **(1 + i)^5 = -4 - 4i

Conclusion

Therefore, using DeMoivre's Theorem, we have determined that (1 + i)^5 = -4 - 4i. This process showcases the utility of complex number representation and DeMoivre's Theorem in efficiently calculating powers of complex numbers.