%5E6.jpeg "(2+2i)^6")
Exploring the Power of Complex Numbers: (2 + 2i)^6
The world of complex numbers extends beyond the familiar realm of real numbers, offering intriguing possibilities. One such exploration involves raising a complex number to a power, as in the case of (2 + 2i)^6. This article delves into the process of calculating this power and unveils the fascinating results.
De Moivre's Theorem: A Powerful Tool
To tackle (2 + 2i)^6, we can utilize De Moivre's theorem, a fundamental principle in complex number manipulation. It 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 effectively transforms the process of raising a complex number to a power into a series of straightforward multiplications and trigonometric calculations.
Applying De Moivre's Theorem
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Polar Form: We begin by converting (2 + 2i) into polar form. The modulus, r, is calculated as:
r = √(2^2 + 2^2) = √8 = 2√2
The argument, θ, is determined by considering the quadrant of the complex number in the complex plane:
θ = arctan(2/2) = π/4
Therefore, (2 + 2i) in polar form is: 2√2(cos π/4 + i sin π/4).
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Applying the Theorem: Now, we apply De Moivre's theorem to (2 + 2i)^6:
(2√2(cos π/4 + i sin π/4))^6 = (2√2)^6(cos (6 * π/4) + i sin (6 * π/4))
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Simplifying: Simplifying the expression, we get:
512(cos (3π/2) + i sin (3π/2))
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Converting back to Rectangular Form: Finally, we convert back to rectangular form:
512(0 + i * (-1)) = -512i
Conclusion
Through the application of De Moivre's theorem, we've successfully calculated (2 + 2i)^6, revealing the result to be -512i. This exploration highlights the elegance and utility of complex number operations, showcasing their ability to simplify seemingly complex calculations. By understanding and applying fundamental principles like De Moivre's theorem, we can unlock the intricacies and beauty of the complex number system.