If `z_(i)` (where `i=1, 2,………………..6`) be the roots of the equation `z^(6)+z^(4)-2=0`, then `Sigma_(i=1)^(6)|z_(i)|^(4)` is equal to

Options

A. 4
B. 6
C. 8
D. 10

Answer

To solve the equation \( z^6 + z^4 - 2 = 0 \) and find the sum \( \Sigma_{i=1}^{6} |z_i|^4 \), we will follow these steps:

### Step 1: Factor the equation
We start with the equation:
\[ z^6 + z^4 - 2 = 0 \]
We can rewrite this as:
\[ z^6 - 1 + z^4 - 1 = 0 \]
This can be factored using the difference of cubes and squares:
\[ (z^2 - 1)(z^4 + 2z^2 + 2) = 0 \]

### Step 2: Solve the first factor
Setting the first factor to zero:
\[ z^2 - 1 = 0 \]
This gives us:
\[ z = 1 \quad \text{and} \quad z = -1 \]

### Step 3: Solve the second factor
Now, we solve the second factor:
\[ z^4 + 2z^2 + 2 = 0 \]
Let \( t = z^2 \). Then we have:
\[ t^2 + 2t + 2 = 0 \]
Using the quadratic formula:
\[ t = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 - 8}}{2} = \frac{-2 \pm \sqrt{-4}}{2} = -1 \pm i \]
Thus, the roots for \( z^2 \) are:
\[ z^2 = -1 + i \quad \text{and} \quad z^2 = -1 - i \]

### Step 4: Find the roots \( z \)
Taking square roots, we find:
1. For \( z^2 = -1 + i \):
\[ z = \sqrt{-1 + i} \quad \text{and} \quad z = -\sqrt{-1 + i} \]
2. For \( z^2 = -1 - i \):
\[ z = \sqrt{-1 - i} \quad \text{and} \quad z = -\sqrt{-1 - i} \]

### Step 5: Calculate the magnitudes
Now we need to calculate \( |z|^4 \) for all roots:
1. For \( z = 1 \) and \( z = -1 \):
\[ |z_1|^4 = 1^4 = 1 \quad \text{and} \quad |z_2|^4 = (-1)^4 = 1 \]

2. For \( z^2 = -1 + i \):
\[ |z^2| = \sqrt{(-1)^2 + 1^2} = \sqrt{2} \implies |z| = \sqrt[4]{2} \implies |z|^4 = 2 \]
This applies to both roots derived from \( -1 + i \).

3. For \( z^2 = -1 - i \):
\[ |z^2| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2} \implies |z| = \sqrt[4]{2} \implies |z|^4 = 2 \]
This applies to both roots derived from \( -1 - i \).

### Step 6: Summation of magnitudes
Now we can sum the magnitudes:
\[ \Sigma_{i=1}^{6} |z_i|^4 = |z_1|^4 + |z_2|^4 + |z_3|^4 + |z_4|^4 + |z_5|^4 + |z_6|^4 = 1 + 1 + 2 + 2 + 2 + 2 = 10 \]

### Final Answer
Thus, the value of \( \Sigma_{i=1}^{6} |z_i|^4 \) is:
\[ \boxed{10} \]