Use de Moivre's Theorem to find the following. Write your answer in standard form.(3 − i)4

1. Break Down the Problem

To find $(3 - i)^4$ using de Moivre's Theorem, we first need to express the complex number in polar form.

2. Relevant Concepts

The polar form of a complex number $z = a + bi$ is given by:

• Magnitude: $r = \sqrt{a^2 + b^2}$
• Angle (argument): $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

Then, de Moivre's Theorem states that: $z^n = r^n (\cos(n\theta) + i\sin(n\theta))$

3. Analysis and Detail

1. Calculate the magnitude $r$: $r = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}$

2. Calculate the argument $\theta$: $\theta = \tan^{-1}\left(\frac{-1}{3}\right)$

3. Using de Moivre's Theorem:

   • We can express $(3 - i)$ in polar form: $(3 - i) = \sqrt{10} \left( \cos\left(\tan^{-1}\left(-\frac{1}{3}\right)\right) + i \sin\left(\tan^{-1}\left(-\frac{1}{3}\right)\right) \right)$

   • Now, using $n = 4$: $(3 - i)^4 = r^4 \left( \cos(4\theta) + i\sin(4\theta) \right)$

   • Compute $r^4$: $r^4 = (\sqrt{10})^4 = 10^2 = 100$

4. Calculate $4\theta$

Using the angle $\theta = \tan^{-1}\left(-\frac{1}{3}\right)$, we find $4\theta$: $\theta \approx -0.3217 \text{ radians (approximately)}$ Thus, $4\theta \approx -1.2868 \text{ radians}$

5. Calculate $\cos(4\theta)$ and $\sin(4\theta)$

• Using approximate values: $\cos(-1.2868) \approx 0.2806, \quad \sin(-1.2868) \approx -0.9602$

6. Combine Results

Using these values: $(3 - i)^4 = 100 \left( 0.2806 - 0.9602i \right) \approx 28.06 - 96.02i$

Final Answer

$(3 - i)^4 \approx 28.06 - 96.02i$