Evaluate: `" "(i)" "i^(19)" "(ii)" "i^(62)" "(iii)" "i^(373)`

Answer

To evaluate the powers of the imaginary unit \( i \), we can use the fact that \( i^4 = 1 \). This means that the powers of \( i \) repeat every four terms:

- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

To find \( i^n \) for any integer \( n \), we can reduce \( n \) modulo 4.

### Step-by-Step Solution:

**(i) Evaluate \( i^{19} \)**

1. **Find \( 19 \mod 4 \)**:
\[ 19 \div 4 = 4 \quad \text{(remainder 3)} \]
Therefore, \( 19 \mod 4 = 3 \).

2. **Use the result**:
\[ i^{19} = i^3 \]

3. **Determine \( i^3 \)**:
\[ i^3 = -i \]

Thus, \( i^{19} = -i \).

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**(ii) Evaluate \( i^{62} \)**

1. **Find \( 62 \mod 4 \)**:
\[ 62 \div 4 = 15 \quad \text{(remainder 2)} \]
Therefore, \( 62 \mod 4 = 2 \).

2. **Use the result**:
\[ i^{62} = i^2 \]

3. **Determine \( i^2 \)**:
\[ i^2 = -1 \]

Thus, \( i^{62} = -1 \).

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**(iii) Evaluate \( i^{373} \)**

1. **Find \( 373 \mod 4 \)**:
\[ 373 \div 4 = 93 \quad \text{(remainder 1)} \]
Therefore, \( 373 \mod 4 = 1 \).

2. **Use the result**:
\[ i^{373} = i^1 \]

3. **Determine \( i^1 \)**:
\[ i^1 = i \]

Thus, \( i^{373} = i \).

### Final Answers:
- \( i^{19} = -i \)
- \( i^{62} = -1 \)
- \( i^{373} = i \)

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