To solve the equation $z^4 = (8 - 6i)^2$, let's go step by step.

Step 1: Simplify the right-hand side

We need to simplify the expression $(8 - 6i)^2$.

Using the formula for the square of a binomial: $(a - b)^2 = a^2 - 2ab + b^2$ Let $a = 8$ and $b = 6i$: $(8 - 6i)^2 = 8^2 - 2 \cdot 8 \cdot 6i + (6i)^2$

First, compute the individual parts: $8^2 = 64$ $-2 \cdot 8 \cdot 6i = -96i$ $(6i)^2 = -36$

Now, put it all together: $(8 - 6i)^2 = 64 - 96i - 36 = 28 - 96i$

Step 2: Rewrite the equation

Now the equation becomes: $z^4 = 28 - 96i$

Step 3: Convert the right-hand side to polar form

We want to convert $28 - 96i$ to polar form, as this will make finding the fourth roots easier.

First, calculate the magnitude (modulus) $r$: $r = \sqrt{28^2 + (-96)^2} = \sqrt{784 + 9216} = \sqrt{10000} = 100$

Next, calculate the argument $\theta$. We use the formula: $\theta = \text{tan}^{-1} \left( \frac{-96}{28} \right)$ $\theta = \text{tan}^{-1} \left( -\frac{96}{28} \right) = \text{tan}^{-1} \left( -\frac{24}{7} \right)$

Using a calculator: $\theta \approx -1.2925 \, \text{radians}$

So, in polar form, we have: $28 - 96i = 100 \text{cis}(-1.2925)$ where $\text{cis}(\theta)$ is shorthand for $\cos(\theta) + i\sin(\theta)$.

Step 4: Take the fourth root

Now, we solve for $z$. The equation is $z^4 = 100 \text{cis}(-1.2925)$.

To find the fourth roots, we take the fourth root of the magnitude and divide the argument by 4. The magnitude of $z$ will be: $|z| = \sqrt[4]{100} = 100^{1/4} = \sqrt{10}$

The argument for $z$ will be: $\frac{-1.2925 + 2k\pi}{4} \quad \text{for} \, k = 0, 1, 2, 3$

Thus, the four solutions are: $z_k = \sqrt{10} \, \text{cis} \left( \frac{-1.2925 + 2k\pi}{4} \right) \quad \text{for} \, k = 0, 1, 2, 3$

These are the four values of $z$.

Would you like more details on finding the exact angles or any part of this solution?

Here are some related questions:

1. How do you convert complex numbers to polar form?
2. What are the steps to compute the argument (angle) of a complex number?
3. How do you find the roots of a complex number using De Moivre's Theorem?
4. How do you handle negative arguments when solving roots of complex numbers?
5. Can you express complex numbers in terms of Euler's formula?

Tip: When solving complex equations, converting numbers into polar form often simplifies root extraction and angle calculations.