Find the value of `i^(2)+(-i)^(4)-i^(6)`.

Answer

To solve the expression \( i^2 + (-i)^4 - i^6 \), we will calculate each term step by step.

### Step 1: Calculate \( i^2 \)
We know that:
\[ i = \sqrt{-1} \]
Thus,
\[ i^2 = -1 \]

### Step 2: Calculate \( (-i)^4 \)
First, we can express \( -i \) as:
\[ -i = -1 \cdot i \]
Now, we can calculate \( (-i)^4 \):
\[ (-i)^4 = ((-1) \cdot i)^4 = (-1)^4 \cdot i^4 \]
Since \( (-1)^4 = 1 \) and \( i^4 = 1 \) (because \( i^4 = (i^2)^2 = (-1)^2 = 1 \)), we have:
\[ (-i)^4 = 1 \cdot 1 = 1 \]

### Step 3: Calculate \( -i^6 \)
Next, we calculate \( i^6 \):
\[ i^6 = i^{4+2} = i^4 \cdot i^2 \]
From our previous calculations, we know \( i^4 = 1 \) and \( i^2 = -1 \):
\[ i^6 = 1 \cdot (-1) = -1 \]
Thus,
\[ -i^6 = -(-1) = 1 \]

### Step 4: Combine the results
Now we can combine all the calculated values:
\[ i^2 + (-i)^4 - i^6 = -1 + 1 - (-1) \]
Substituting the values we found:
\[ = -1 + 1 + 1 \]
\[ = 1 \]

### Final Answer
The value of \( i^2 + (-i)^4 - i^6 \) is \( \boxed{1} \).

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