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Exploring the Power of Complex Numbers: (-1 + i)^5
The expression (-1 + i)^5 may seem intimidating at first glance, but with the right approach, we can simplify it quite easily. This article will guide you through the process, breaking down the solution into manageable steps.
Understanding Complex Numbers
Before we delve into the calculation, it's crucial to understand what complex numbers are and how they behave. A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.
The Power of De Moivre's Theorem
De Moivre's Theorem provides a powerful tool for simplifying powers of complex numbers. 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))
Applying De Moivre's Theorem to (-1 + i)^5
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Convert to Polar Form: First, we need to convert (-1 + i) into polar form. To do this, we find its magnitude (r) and its angle (θ).
- Magnitude: r = √((-1)^2 + 1^2) = √2
- Angle: θ = arctan(1/-1) = -45° (or 315° in the principal argument range)
Therefore, (-1 + i) = √2 (cos(-45°) + i sin(-45°))
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Apply De Moivre's Theorem: Now, we apply De Moivre's Theorem to find (-1 + i)^5:
(-1 + i)^5 = (√2)^5 (cos(-45° * 5) + i sin(-45° * 5))
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Simplify:
(-1 + i)^5 = 4√2 (cos(-225°) + i sin(-225°))
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Convert back to Rectangular Form: Finally, we convert the polar form back to rectangular form. Since cos(-225°) = -√2/2 and sin(-225°) = √2/2, we get:
(-1 + i)^5 = 4√2 * (-√2/2 + i √2/2) = -4 + 4i
Conclusion
By leveraging De Moivre's Theorem, we effectively calculated (-1 + i)^5 and obtained the simplified result of -4 + 4i. This demonstrates the power of using polar forms and complex number identities to manipulate and solve seemingly complex problems.