Question

What are the polar forms of the complex fourth roots of the number 4−43√i?

Answer 1

To find the polar forms of the complex fourth roots of the number \( 4 - 43\sqrt{i} \), we will first convert \( 4 - 43\sqrt{i} \) into its polar form, and then find the fourth roots.

### Step 1: Convert \( \sqrt{i} \) into polar form.

The complex number \( i \) can be expressed in polar form as follows:
\[
i = e^{i\frac{\pi}{2}}.
\]
Hence,
\[
\sqrt{i} = \sqrt{e^{i\frac{\pi}{2}}} = e^{i\frac{\pi}{4}}.
\]

### Step 2: Calculate \( -43\sqrt{i} \).

Now we can express \( -43\sqrt{i} \):
\[
-43\sqrt{i} = -43e^{i\frac{\pi}{4}} = 43e^{i(\frac{\pi}{4} + \pi)} = 43e^{i\frac{5\pi}{4}}.
\]

### Step 3: Combine with the real part.

Now we can express \( 4 - 43\sqrt{i} \):
\[
4 - 43\sqrt{i} = 4 + 43e^{i\frac{5\pi}{4}}.
\]
To add these two parts, we have the real part \( 4 \) and the point represented by \( 43e^{i\frac{5\pi}{4}} \).

### Step 4: Convert \( 43e^{i\frac{5\pi}{4}} \) to rectangular form.

The rectangular form of \( 43e^{i\frac{5\pi}{4}} \) is:
\[
43\left(\cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right)\right) = 43\left(-\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\right) = -\frac{43\sqrt{2}}{2} - i\frac{43\sqrt{2}}{2}.
\]

### Step 5: Sum the rectangular form.

Now we compute \( 4 - 43\sqrt{i} \):
\[
4 - 43\sqrt{i} = 4 + \left(-\frac{43\sqrt{2}}{2}\right) - i\frac{43\sqrt{2}}{2}.
\]
Let's combine the real parts:
\[
4 - \frac{43\sqrt{2}}{2} - i\frac{43\sqrt{2}}{2}.
\]

### Step 6: Find magnitude and argument.

Let \( x = 4 - \frac{43\sqrt{2}}{2} \) and \( y = -\frac{43\sqrt{2}}{2} \).

1. The magnitude \( r \) is given by:
\[
r = \sqrt{x^2 + y^2}.
\]

2. The argument \( \theta \) is given by:
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right).
\]

### Step 7: Calculate \( r \) and \( \theta \).

To compute \( x \) accurately, we'll note that \( 43\sqrt{2} \approx 43 \times 1.414 = 60.62 \). Thus,
\[
x \approx 4 - 30.31 = -26.31,
\]
and \( y = -30.31 \). Thus,
\[
r = \sqrt{(-26.31)^2 + (-30.31)^2} \approx \sqrt{691.5961 + 919.0561} \approx \sqrt{1610.6522} \approx 40.12.
\]
The angle \( \theta \) will use
\[
\theta = \tan^{-1}\left(\frac{-30.31}{-26.31}\right)
\]
which is in the third quadrant.

### Step 8: Find fourth roots.

The fourth roots can be computed as:
\[
z_k = \sqrt[4]{r} \left( \cos\left(\frac{\theta + 2\pi k}{4}\right) + i\sin\left(\frac{\theta + 2\pi k}{4}\right) \right),
\]
for \( k = 0, 1, 2, 3 \).

### Conclusion

Calculate numerical values:
1. \( r^{1/4} \approx 40.12^{1/4} = \sqrt{\sqrt{40.12}} \approx 3.16 \).
2. Calculate \( \theta = \tan^{-1}\left(\frac{-30.31}{-26.31}\right) \), converting to radians and normalizing within \( [0, 2\pi] \).

The resulting polar forms \( z_k \) can be calculated and expressed for \( k = 0, 1, 2, 3 \).

To find the specific fourth roots numerically or symbolically, you'll need numerical assistance for increased precision, especially for angle calculations and the complex exponentials derived here.