%5E4.jpeg "(1+2i)^4")
Exploring the Power of Complex Numbers: (1+2i)^4
In the realm of mathematics, complex numbers often present intriguing challenges, especially when raised to higher powers. Let's delve into the intriguing case of (1 + 2i)^4, where 'i' represents the imaginary unit (√-1).
De Moivre's Theorem: Our Key to Success
To tackle this calculation efficiently, we'll employ De Moivre's Theorem, a powerful tool for dealing with complex numbers raised to powers. The theorem states:
(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)
Where:
- n is any integer
- θ is the angle of the complex number in polar form
Breaking Down the Problem
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Convert to Polar Form: First, we need to express (1 + 2i) in its polar form. We can achieve this by:
- Finding the magnitude (or modulus) |1 + 2i| = √(1² + 2²) = √5.
- Determining the angle θ using arctan(2/1) ≈ 63.43°.
Therefore, (1 + 2i) in polar form is √5 (cos 63.43° + i sin 63.43°).
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Applying De Moivre's Theorem: Now, we can apply the theorem: (1 + 2i)^4 = (√5 (cos 63.43° + i sin 63.43°))^4 = 5² (cos (4 * 63.43°) + i sin (4 * 63.43°)) = 25 (cos 253.72° + i sin 253.72°)
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Converting Back to Rectangular Form: Finally, we convert the result back to the standard rectangular form (a + bi): 25 (cos 253.72° + i sin 253.72°) ≈ 25 (-0.31 - 0.95i) = -7.75 - 23.75i
Conclusion
By leveraging De Moivre's Theorem, we successfully calculated (1 + 2i)^4, obtaining the result -7.75 - 23.75i. This exploration demonstrates the power and elegance of complex numbers and their application in mathematical calculations.