Let `Z=re^(itheta)(r gt 0 and pi lt theta lt 3pi)` is a root of the equation `Z^(8)-Z^(7)+Z^(6)-Z^(5)+Z^(4)-Z^(3)+Z^(2)-Z+1=0`.
the sum of all values of `theta` is `kpi`. Then k is equal to

Answer

To solve the given problem, we need to analyze the equation:

\[ Z^8 - Z^7 + Z^6 - Z^5 + Z^4 - Z^3 + Z^2 - Z + 1 = 0 \]

where \( Z = re^{i\theta} \) and \( r > 0 \) with \( \pi < \theta < 3\pi \).

### Step 1: Factor the Polynomial
We can factor the polynomial by multiplying it by \( Z + 1 \):

\[ (Z + 1)(Z^8 - Z^7 + Z^6 - Z^5 + Z^4 - Z^3 + Z^2 - Z + 1) = 0 \]

This gives us:

\[ Z^9 + 1 = 0 \]

The roots of this equation are the 9th roots of -1.

### Step 2: Find the Roots
The 9th roots of -1 can be expressed as:

\[ Z_k = e^{i\left(\frac{\pi + 2k\pi}{9}\right)} \quad \text{for } k = 0, 1, 2, \ldots, 8 \]

### Step 3: Calculate the Angles
The angles corresponding to these roots are:

\[ \theta_k = \frac{\pi + 2k\pi}{9} \]

### Step 4: Determine Valid Angles
We need to find which of these angles fall within the range \( \pi < \theta < 3\pi \).

Calculating the angles for \( k = 0, 1, 2, \ldots, 8 \):

- For \( k = 0 \):
\[ \theta_0 = \frac{\pi}{9} \]
- For \( k = 1 \):
\[ \theta_1 = \frac{3\pi}{9} = \frac{\pi}{3} \]
- For \( k = 2 \):
\[ \theta_2 = \frac{5\pi}{9} \]
- For \( k = 3 \):
\[ \theta_3 = \frac{7\pi}{9} \]
- For \( k = 4 \):
\[ \theta_4 = \frac{9\pi}{9} = \pi \]
- For \( k = 5 \):
\[ \theta_5 = \frac{11\pi}{9} \]
- For \( k = 6 \):
\[ \theta_6 = \frac{13\pi}{9} \]
- For \( k = 7 \):
\[ \theta_7 = \frac{15\pi}{9} = \frac{5\pi}{3} \]
- For \( k = 8 \):
\[ \theta_8 = \frac{17\pi}{9} \]

### Step 5: Select Valid Angles
From the above calculations, the valid angles that fall within \( \pi < \theta < 3\pi \) are:

- \( \theta_5 = \frac{11\pi}{9} \)
- \( \theta_6 = \frac{13\pi}{9} \)
- \( \theta_7 = \frac{15\pi}{9} = \frac{5\pi}{3} \)
- \( \theta_8 = \frac{17\pi}{9} \)

### Step 6: Sum of Valid Angles
Now, we sum these valid angles:

\[ \text{Sum} = \frac{11\pi}{9} + \frac{13\pi}{9} + \frac{15\pi}{9} + \frac{17\pi}{9} = \frac{56\pi}{9} \]

### Step 7: Express in Terms of \( k\pi \)
We need to express this sum in the form \( k\pi \):

\[ \frac{56\pi}{9} = k\pi \implies k = \frac{56}{9} \]

### Final Answer
Thus, the value of \( k \) is:

\[ \boxed{\frac{56}{9}} \]