In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (√2 − i)⁴

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Express the complex number \( \sqrt{2} - i \) in polar form. To do this, find the modulus \( r \) using \( r = \sqrt{(\sqrt{2})^2 + (-1)^2} \) and the argument \( \theta \) using \( \theta = \tan^{-1} \left( \frac{-1}{\sqrt{2}} \right) \).

Write the complex number in polar form as \( r (\cos \theta + i \sin \theta) \).

Apply DeMoivre's Theorem to raise the complex number to the 4th power: \( (r (\cos \theta + i \sin \theta))^4 = r^4 (\cos 4\theta + i \sin 4\theta) \).

Calculate \( r^4 \) and multiply the argument \( \theta \) by 4 to find \( 4\theta \).

Convert the result back to rectangular form by evaluating \( r^4 \cos 4\theta \) for the real part and \( r^4 \sin 4\theta \) for the imaginary part, giving the final answer in the form \( a + bi \).

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주요 개념

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DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number expressed in polar form as r(cos θ + i sin θ), its nth power is r^n (cos nθ + i sin nθ). This theorem simplifies raising complex numbers to powers by working with their magnitude and angle instead of expanding binomials.

Conversion Between Rectangular and Polar Forms

Complex numbers can be represented in rectangular form (a + bi) or polar form (r(cos θ + i sin θ)). Converting involves finding the magnitude r = √(a² + b²) and the argument θ = arctan(b/a). This conversion is essential for applying DeMoivre's Theorem effectively.

Rectangular Form of Complex Numbers

Rectangular form expresses complex numbers as a + bi, where a is the real part and b is the imaginary part. After using DeMoivre's Theorem in polar form, the result is converted back to rectangular form by evaluating r^n cos nθ and r^n sin nθ to find the real and imaginary components.