Let ` alpha ` be the non-real 5 th root of unity. If ` z_1 and z_2 ` are two complex numbers lying on `|z| = 2`, then the value of ` sum_(t=0) ^(4) |z_1 + alpha ^(t)z_2 |^(2) ` is ______ .

Explanation

To solve the problem, we need to calculate the value of the expression:

\[ \sum_{t=0}^{4} |z_1 + \alpha^t z_2|^2 \]

where \(\alpha\) is a non-real fifth root of unity, and \(z_1\) and \(z_2\) are complex numbers with \(|z_1| = |z_2| = 2\).

### Step 1: Understanding the Roots of Unity
The non-real fifth roots of unity are given by:

\[ \alpha = e^{2\pi i / 5}, \quad \alpha^2 = e^{4\pi i / 5}, \quad \alpha^3 = e^{6\pi i / 5}, \quad \alpha^4 = e^{8\pi i / 5} \]

These roots satisfy the equation:

\[ 1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 = 0 \]

### Step 2: Expanding the Expression
We can expand the expression \(|z_1 + \alpha^t z_2|^2\):

\[ |z_1 + \alpha^t z_2|^2 = (z_1 + \alpha^t z_2)(\overline{z_1 + \alpha^t z_2}) = |z_1|^2 + |\alpha^t z_2|^2 + z_1 \overline{\alpha^t z_2} + \overline{z_1} \alpha^t z_2 \]

Since \(|\alpha^t z_2| = |z_2| = 2\), we have:

\[ |z_1 + \alpha^t z_2|^2 = |z_1|^2 + |z_2|^2 + z_1 \overline{\alpha^t z_2} + \overline{z_1} \alpha^t z_2 \]

### Step 3: Substitute the Moduli
Given that \(|z_1| = |z_2| = 2\):

\[ |z_1|^2 = 4, \quad |z_2|^2 = 4 \]

Thus, we can rewrite:

\[ |z_1 + \alpha^t z_2|^2 = 4 + 4 + z_1 \overline{\alpha^t z_2} + \overline{z_1} \alpha^t z_2 = 8 + z_1 \overline{\alpha^t z_2} + \overline{z_1} \alpha^t z_2 \]

### Step 4: Summing Over \(t\)
Now we need to sum this expression from \(t=0\) to \(t=4\):

\[ \sum_{t=0}^{4} |z_1 + \alpha^t z_2|^2 = \sum_{t=0}^{4} \left(8 + z_1 \overline{\alpha^t z_2} + \overline{z_1} \alpha^t z_2\right) \]

This simplifies to:

\[ \sum_{t=0}^{4} |z_1 + \alpha^t z_2|^2 = 5 \cdot 8 + \sum_{t=0}^{4} (z_1 \overline{\alpha^t z_2} + \overline{z_1} \alpha^t z_2) \]

### Step 5: Evaluating the Summation
The term \(\sum_{t=0}^{4} \alpha^t\) is the sum of the fifth roots of unity, which equals zero:

\[ \sum_{t=0}^{4} \alpha^t = 0 \]

Thus, we have:

\[ \sum_{t=0}^{4} (z_1 \overline{\alpha^t z_2} + \overline{z_1} \alpha^t z_2) = z_1 \cdot 0 + \overline{z_1} \cdot 0 = 0 \]

### Final Calculation
Putting it all together:

\[ \sum_{t=0}^{4} |z_1 + \alpha^t z_2|^2 = 5 \cdot 8 + 0 = 40 \]

Thus, the final answer is:

\[ \boxed{40} \]