Evaluate the following
(i) `i^(9)`
(ii) `i^(342)`
(iii) `i^(998)`
(iv) `i^(-63)`
(v) `(i^(3)+(1)/(i^(3)))`

Answer

To evaluate the given expressions involving the imaginary unit \( i \) (where \( i = \sqrt{-1} \)), we can use the cyclical nature of powers of \( i \). The powers of \( i \) cycle every four terms as follows:

- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \) (and so on)

Using this pattern, we can evaluate each part step by step.

### Part (i): Evaluate \( i^9 \)

1. **Find the remainder when 9 is divided by 4**:
\[ 9 \div 4 = 2 \quad \text{(remainder 1)} \]
2. **Use the cyclical pattern**:
\[ i^9 = i^{4 \cdot 2 + 1} = (i^4)^2 \cdot i^1 = 1^2 \cdot i = i \]

**Answer**: \( i^9 = i \)

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### Part (ii): Evaluate \( i^{342} \)

1. **Find the remainder when 342 is divided by 4**:
\[ 342 \div 4 = 85 \quad \text{(remainder 2)} \]
2. **Use the cyclical pattern**:
\[ i^{342} = i^{4 \cdot 85 + 2} = (i^4)^{85} \cdot i^2 = 1^{85} \cdot (-1) = -1 \]

**Answer**: \( i^{342} = -1 \)

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### Part (iii): Evaluate \( i^{998} \)

1. **Find the remainder when 998 is divided by 4**:
\[ 998 \div 4 = 249 \quad \text{(remainder 2)} \]
2. **Use the cyclical pattern**:
\[ i^{998} = i^{4 \cdot 249 + 2} = (i^4)^{249} \cdot i^2 = 1^{249} \cdot (-1) = -1 \]

**Answer**: \( i^{998} = -1 \)

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### Part (iv): Evaluate \( i^{-63} \)

1. **Convert to positive exponent**:
\[ i^{-63} = \frac{1}{i^{63}} \]
2. **Find the remainder when 63 is divided by 4**:
\[ 63 \div 4 = 15 \quad \text{(remainder 3)} \]
3. **Use the cyclical pattern**:
\[ i^{63} = i^{4 \cdot 15 + 3} = (i^4)^{15} \cdot i^3 = 1^{15} \cdot (-i) = -i \]
4. **Substitute back**:
\[ i^{-63} = \frac{1}{-i} = -\frac{1}{i} \cdot \frac{i}{i} = -\frac{i}{-1} = i \]

**Answer**: \( i^{-63} = i \)

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### Part (v): Evaluate \( i^3 + \frac{1}{i^3} \)

1. **Calculate \( i^3 \)**:
\[ i^3 = -i \]
2. **Calculate \( \frac{1}{i^3} \)**:
\[ \frac{1}{i^3} = \frac{1}{-i} = -\frac{1}{i} \cdot \frac{i}{i} = -\frac{i}{-1} = i \]
3. **Combine the results**:
\[ i^3 + \frac{1}{i^3} = -i + i = 0 \]

**Answer**: \( i^3 + \frac{1}{i^3} = 0 \)

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### Summary of Answers:
1. \( i^9 = i \)
2. \( i^{342} = -1 \)
3. \( i^{998} = -1 \)
4. \( i^{-63} = i \)
5. \( i^3 + \frac{1}{i^3} = 0 \)

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