Simplify the following :
(i) `i^(97)`
(ii) `i^(8)`
(iii) `(1)/(i^(3))`
(iv) `(-i)^(14)`
(v) `i^(-22)`
(vi) `i^(-63)`

Explanation

Let's simplify each part step by step:

### (i) Simplify `i^(97)`

1. **Rewrite the exponent**:
\[ i^{97} = i^{96 + 1} = i^{96} \cdot i^1 \]

2. **Use the property of powers of i**:
\[ i^{96} = (i^4)^{24} = 1^{24} = 1 \]
(since \(i^4 = 1\))

3. **Combine the results**:
\[ i^{97} = 1 \cdot i = i \]

### Answer for (i):
\[ i^{97} = i \]

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### (ii) Simplify `i^(8)`

1. **Rewrite the exponent**:
\[ i^8 = (i^4)^2 \]

2. **Use the property of powers of i**:
\[ i^4 = 1 \implies i^8 = 1^2 = 1 \]

### Answer for (ii):
\[ i^8 = 1 \]

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### (iii) Simplify `(1)/(i^(3))`

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

2. **Rewrite the expression**:
\[ \frac{1}{i^3} = \frac{1}{-i} \]

3. **Rationalize the denominator**:
\[ \frac{1}{-i} \cdot \frac{i}{i} = \frac{i}{-i^2} = \frac{i}{-(-1)} = \frac{i}{1} = i \]

### Answer for (iii):
\[ \frac{1}{i^3} = i \]

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### (iv) Simplify `(-i)^(14)`

1. **Rewrite the expression**:
\[ (-i)^{14} = (-1)^{14} \cdot i^{14} \]

2. **Calculate each part**:
\[ (-1)^{14} = 1 \quad \text{and} \quad i^{14} = (i^2)^7 = (-1)^7 = -1 \]

3. **Combine the results**:
\[ (-i)^{14} = 1 \cdot (-1) = -1 \]

### Answer for (iv):
\[ (-i)^{14} = -1 \]

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### (v) Simplify `i^(-22)`

1. **Rewrite the exponent**:
\[ i^{-22} = \frac{1}{i^{22}} \]

2. **Calculate \(i^{22}\)**:
\[ i^{22} = (i^4)^5 \cdot i^2 = 1^5 \cdot (-1) = -1 \]

3. **Combine the results**:
\[ i^{-22} = \frac{1}{-1} = -1 \]

### Answer for (v):
\[ i^{-22} = -1 \]

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### (vi) Simplify `i^(-63)`

1. **Rewrite the exponent**:
\[ i^{-63} = \frac{1}{i^{63}} \]

2. **Calculate \(i^{63}\)**:
\[ i^{63} = i^{62 + 1} = i^{62} \cdot i = (i^4)^{15} \cdot i^2 = 1^{15} \cdot (-1) = -1 \]

3. **Combine the results**:
\[ i^{-63} = \frac{1}{-1} = -1 \]

### Answer for (vi):
\[ i^{-63} = -1 \]

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### Summary of Answers:
1. \(i^{97} = i\)
2. \(i^8 = 1\)
3. \(\frac{1}{i^3} = i\)
4. \((-i)^{14} = -1\)
5. \(i^{-22} = -1\)
6. \(i^{-63} = -1\)

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