In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (1 − i)⁵

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1. Express the complex number in polar form. The complex number is $1 - i$, which can be written as $z = 1 - i$. Find the modulus $r = \sqrt{1^2 + (-1)^2} = \sqrt{2}$ and the argument $\theta = \tan^{-1}\left(\frac{-1}{1}\right) = -\frac{\pi}{4}$. So, $z = \sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))$.
2. Apply DeMoivre's Theorem: $(r(\cos \theta + i\sin \theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))$. Here, $n = 5$, $r = \sqrt{2}$, and $\theta = -\frac{\pi}{4}$.
3. Calculate $r^n$: $(\sqrt{2})^5 = (2^{1/2})^5 = 2^{5/2} = 4\sqrt{2}$.
4. Calculate $n\theta$: $5 \times (-\frac{\pi}{4}) = -\frac{5\pi}{4}$. Substitute into DeMoivre's Theorem: $4\sqrt{2}(\cos(-\frac{5\pi}{4}) + i\sin(-\frac{5\pi}{4}))$.
5. Convert back to rectangular form: Use the identities $\cos(-\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}$ and $\sin(-\frac{5\pi}{4}) = \frac{1}{\sqrt{2}}$. Substitute these into the expression to get $4\sqrt{2}(-\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}})$ and simplify.

Key Concepts

DeMoivre's Theorem

DeMoivre's Theorem states that for any complex number in polar form, z = r(cos θ + i sin θ), the nth power of z can be expressed as z^n = r^n (cos(nθ) + i sin(nθ)). This theorem simplifies the process of raising complex numbers to a power by converting them to polar coordinates, making calculations more manageable.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be represented in two forms: rectangular form, a + bi, where a is the real part and b is the imaginary part, and polar form, r(cos θ + i sin θ), where r is the magnitude and θ is the angle. Understanding how to convert between these forms is essential for applying DeMoivre's Theorem effectively.

Magnitude and Argument of a Complex Number

The magnitude of a complex number z = a + bi is given by |z| = √(a² + b²), representing its distance from the origin in the complex plane. The argument, θ, is the angle formed with the positive real axis, calculated using θ = arctan(b/a). These values are crucial for converting a complex number to polar form before applying DeMoivre's Theorem.