VIPSolutionsfind the Indicated roots and graph them In the complex plane. we » we I we » Is ES N we FI EE Ea FR Era ER Ea ER Era ER we Sw ) wy w wool @0 @0 @0 @ Rafrin th bos rom th graph yo slctd bo, gv textos. C1, ne x 2 . 2
Question:
find the Indicated roots and graph them In the complex plane.
we » we I
we » Is ES
N we FI EE
Ea FR Era ER Ea ER Era ER
we Sw )
wy w wool
@0 @0 @0 @
Rafrin th bos rom th graph yo slctd bo, gv textos.
C1,
ne x
2 .
2
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Asked by: Test User 100047
Created at: 2025-05-10 04:08:33
Asked by: Test User 100047
Created at: 2025-05-10 04:08:33
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Let's solve the problem step by step:
### **Given:**
Find the fourth roots of \(-256i\).
### **Step 1: Express \(-256i\) in Polar Form**
- \( -256i = 0 - 256i \)
- The modulus \( r = 256 \)
- The argument \( \theta \) is \( -\frac{\pi}{2} \) (since it is on the negative imaginary axis)
So,
\[
-256i = 256 \text{cis} \left(-\frac{\pi}{2}\right)
\]
where \(\text{cis}(\theta) = \cos(\theta) + i\sin(\theta)\)
---
### **Step 2: Find the Fourth Roots**
The nth roots of \( r \text{cis}(\theta) \) are:
\[
w_k = \sqrt[4]{256} \text{cis} \left( \frac{-\frac{\pi}{2} + 2k\pi}{4} \right) \quad \text{for} \ k = 0, 1, 2, 3
\]
- \(\sqrt[4]{256} = 4\)
So,
\[
w_k = 4 \text{cis} \left( \frac{-\frac{\pi}{2} + 2k\pi}{4} \right)
\]
Let's calculate each root:
#### \(k = 0\):
\[
w_0 = 4 \text{cis}\left(\frac{-\frac{\pi}{2}}{4}\right) = 4 \text{cis}\left(-\frac{\pi}{8}\right)
\]
#### \(k = 1\):
\[
w_1 = 4 \text{cis}\left(\frac{-\frac{\pi}{2} + 2\pi}{4}\right) = 4 \text{cis}\left(\frac{3\pi}{8}\right)
\]
#### \(k = 2\):
\[
w_2 = 4 \text{cis}\left(\frac{-\frac{\pi}{2} + 4\pi}{4}\right) = 4 \text{cis}\left(\frac{7\pi}{8}\right)
\]
#### \(k = 3\):
\[
w_3 = 4 \text{cis}\left(\frac{-\frac{\pi}{2} + 6\pi}{4}\right) = 4 \text{cis}\left(\frac{11\pi}{8}\right)
\]
---
### **Step 3: Match with the Correct Graph**
Look for the plot where the points are:
- All at radius 4 from the origin
- Angles: \( -\frac{\pi}{8}, \frac{3\pi}{8}, \frac{7\pi}{8}, \frac{11\pi}{8} \)
**The correct graph is the third one from the left.**
---
### **Step 4: Assign the Labels**
Referring to the labels in the selected graph:
- \(w_0\) is at \( -\frac{\pi}{8} \) (just below the positive real axis)
- \(w_1\) is at \( \frac{3\pi}{8} \) (in the first quadrant)
- \(w_2\) is at \( \frac{7\pi}{8} \) (in the second quadrant, near negative real axis)
- \(w_3\) is at \( \frac{11\pi}{8} \) (in the third quadrant)
---
### **Step 5: Give the Exact Roots**
\[
\boxed{
\begin{align*}
w_0 &= 4 \text{cis}\left(-\frac{\pi}{8}\right) \\
w_1 &= 4 \text{cis}\left(\frac{3\pi}{8}\right) \\
w_2 &= 4 \text{cis}\left(\frac{7\pi}{8}\right) \\
w_3 &= 4 \text{cis}\left(\frac{11\pi}{8}\right) \\
\end{align*}
}
\]
If you need them in rectangular form (\(a + bi\)), let me know!
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