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

Verified step by step guidance

1. Express the complex number $\sqrt{2}-i$ in polar form. To do this, find the modulus $r$ using $r=\sqrt{(\sqrt{2}{)}^2+(-1{)}^2}$.
2. Calculate the argument $\theta$ of the complex number using $\theta ={\tan }^{-1}\left(\frac{-1}{\sqrt{2}}\right)$.
3. Apply DeMoivre's Theorem: $(r(\cos \theta +i\sin \theta ){)}^n=r^n(\cos (n\theta )+i\sin (n\theta ))$. Here, $n=4$.
4. Compute $r^4$ and $4\theta$ using the values of $r$ and $\theta$ obtained in the previous steps.
5. Convert the result back to rectangular form by evaluating $r^4(\cos (4\theta )+i\sin (4\theta ))$ to get the real and imaginary components.

Key Concepts

DeMoivre's Theorem

DeMoivre's Theorem states that for any complex number expressed in polar form as r(cos θ + i sin θ), the nth power of the complex number can be calculated as r^n (cos(nθ) + i sin(nθ)). This theorem simplifies the process of raising complex numbers to powers and is essential for solving problems involving complex exponentiation.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be represented in two forms: rectangular form (a + bi, where a and b are real numbers) and polar form (r(cos θ + i sin θ), where r is the modulus and θ is the argument). Understanding how to convert between these forms is crucial for applying DeMoivre's Theorem effectively and for expressing the final answer in the required rectangular form.

Modulus and Argument of a Complex Number

The modulus of a complex number is its distance from the origin in the complex plane, calculated as √(a² + b²), while the argument is the angle formed with the positive real axis, found using the arctan function. These two components are vital for converting a complex number from rectangular to polar form, which is necessary for applying DeMoivre's Theorem.