Simplify : `i^(-6)`

Answer

To simplify \( i^{-6} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ i^{-6} \]
Using the property of exponents, we can rewrite it as:
\[ i^{-6} = \frac{1}{i^{6}} \]

### Step 2: Simplify \( i^{6} \)
Next, we need to find \( i^{6} \). We know that:
\[ i^2 = -1 \]
From this, we can derive the powers of \( i \):
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = i^2 \cdot i = -1 \cdot i = -i \)
- \( i^4 = i^2 \cdot i^2 = (-1) \cdot (-1) = 1 \)

Since \( i^4 = 1 \), the powers of \( i \) repeat every four terms. Thus, we can express \( i^{6} \) as:
\[ i^{6} = i^{4} \cdot i^{2} = 1 \cdot (-1) = -1 \]

### Step 3: Substitute back into the expression
Now we substitute \( i^{6} \) back into our expression:
\[ i^{-6} = \frac{1}{i^{6}} = \frac{1}{-1} = -1 \]

### Final Answer
Thus, the simplified form of \( i^{-6} \) is:
\[ \boxed{-1} \]

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