If `n_1, n_2` are positive integers, then `(1 + i)^(n_1) + ( 1 + i^3)^(n_1) + (1 + i_5)^(n_2) + (1 + i^7)^(n_2)` is real if and only if :

Explanation

To determine when the expression \( (1 + i)^{n_1} + (1 + i^3)^{n_1} + (1 + i^5)^{n_2} + (1 + i^7)^{n_2} \) is real, we will simplify each term step by step.

### Step 1: Simplify \( (1 + i)^{n_1} \)

We can express \( 1 + i \) in polar form:
\[ 1 + i = \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) = \sqrt{2} e^{i \frac{\pi}{4}} \]
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