%5E8.jpeg "(2-2i)^8")
Exploring the Power of Complex Numbers: (2-2i)^8
This article delves into the process of calculating the complex number (2-2i)^8. We'll utilize De Moivre's Theorem, a powerful tool for simplifying complex number exponents.
Understanding De Moivre's Theorem
De Moivre's Theorem states that for any complex number in polar form, z = r(cos θ + i sin θ) and any integer n:
z^n = r^n (cos(nθ) + i sin(nθ))
This theorem allows us to calculate powers of complex numbers efficiently.
Converting to Polar Form
Firstly, we need to express (2-2i) in polar form. We can achieve this by finding the modulus (r) and argument (θ) of the complex number.
- Modulus (r): |2-2i| = √(2² + (-2)²) = √8 = 2√2
- Argument (θ): θ = arctan(-2/2) = -π/4. (We choose -π/4 as the complex number lies in the fourth quadrant)
Therefore, (2-2i) in polar form is 2√2 (cos(-π/4) + i sin(-π/4))
Applying De Moivre's Theorem
Now, we can apply De Moivre's Theorem to calculate (2-2i)^8:
(2-2i)^8 = (2√2)^8 * (cos(-8π/4) + i sin(-8π/4))
Simplifying:
(2-2i)^8 = 256 * (cos(-2π) + i sin(-2π))
Since cos(-2π) = 1 and sin(-2π) = 0, the result is:
(2-2i)^8 = 256
Conclusion
By utilizing De Moivre's Theorem, we successfully calculated (2-2i)^8 and obtained the surprising result of 256. This demonstrates the power of De Moivre's Theorem in simplifying complex number calculations and highlights the fascinating properties of complex numbers.