In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. _ (1−i√3)²

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Identify the complex number given: \(1 - i\sqrt{3}\). We want to find its square, i.e., raise it to the power 2.

Convert the complex number to polar form. First, find the modulus \(r\) using \(r = \sqrt{a^2 + b^2}\), where \(a = 1\) and \(b = -\sqrt{3}\).

Calculate the argument \(\theta\) (angle) using \(\theta = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right)\), making sure to consider the correct quadrant for the complex number.

Apply DeMoivre's Theorem: \((r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta))\), where \(n=2\) in this case.

Convert the result back to rectangular form by calculating \(r^2 \cos(2\theta)\) for the real part and \(r^2 \sin(2\theta)\) for the imaginary part.

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주요 개념

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Complex Numbers in Rectangular and Polar Form

Complex numbers can be expressed in rectangular form as a + bi, where a is the real part and b is the imaginary part. They can also be represented in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). Converting between these forms is essential for applying DeMoivre's Theorem.

DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). This theorem simplifies raising complex numbers to integer powers by working with their magnitude and angle, making it easier to compute powers before converting back to rectangular form.

Conversion from Polar to Rectangular Form

After applying DeMoivre's Theorem, the result is in polar form. To express the answer in rectangular form, use x = r cos θ and y = r sin θ to find the real and imaginary parts. This step is crucial to present the final answer as a + bi, as required by the problem.