Which of the following is equal to `i^(50)+i^(0)`?

Options

A. `1`
B. `2`
C. `-1`
D. `0`

Answer

To solve the expression \( i^{50} + i^{0} \), we can follow these steps:

### Step 1: Evaluate \( i^{0} \)
By the property of exponents, any non-zero number raised to the power of 0 is equal to 1. Therefore:
\[ i^{0} = 1 \]

### Step 2: Evaluate \( i^{50} \)
The powers of \( i \) (the imaginary unit) cycle every four terms:
- \( i^{1} = i \)
- \( i^{2} = -1 \)
- \( i^{3} = -i \)
- \( i^{4} = 1 \)

To find \( i^{50} \), we can reduce the exponent modulo 4:
\[ 50 \mod 4 = 2 \]
This means:
\[ i^{50} = i^{2} = -1 \]

### Step 3: Combine the results
Now we can substitute the values we found back into the original expression:
\[ i^{50} + i^{0} = -1 + 1 \]

### Step 4: Simplify the expression
Simplifying the expression gives:
\[ -1 + 1 = 0 \]

### Final Answer
Thus, the value of \( i^{50} + i^{0} \) is:
\[ \boxed{0} \]

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