(2+2i)^20

Jasonbradley

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Jun 16, 2024

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Calculating (2 + 2i)^20

This article explores the method for calculating the complex number (2 + 2i)^20.

Understanding Complex Numbers

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit defined as the square root of -1 (i.e., i² = -1).

Polar Form of Complex Numbers

To simplify the calculation of powers of complex numbers, it's helpful to convert them to polar form. The polar form of a complex number a + bi is given by r(cos θ + i sin θ), where:

• r is the modulus of the complex number, calculated as √(a² + b²).
• θ is the argument of the complex number, calculated as arctan(b/a) (adjusting the angle based on the quadrant of the complex number).

Applying De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the following holds:

[r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ)

Calculating (2 + 2i)^20

1. Convert to polar form:

   • r = √(2² + 2²) = √8 = 2√2
   • θ = arctan(2/2) = π/4 (since 2 + 2i lies in the first quadrant)
   • Therefore, 2 + 2i = 2√2(cos π/4 + i sin π/4)

2. Apply De Moivre's Theorem:

   • (2√2(cos π/4 + i sin π/4))^20 = (2√2)^20 (cos 20π/4 + i sin 20π/4)

3. Simplify:

   • (2√2)^20 = 2^20 * 2^10 = 2^30
   • cos 20π/4 = cos 5π = -1
   • sin 20π/4 = sin 5π = 0

4. Final result:

   • (2 + 2i)^20 = 2^30 (-1 + 0i) = -2^30

Therefore, (2 + 2i)^20 simplifies to -2^30.