`2i^4 - i^6` = ?

Options

A. i-1
B. i+1
C. `-3`
D. 3

Answer

To solve the expression \(2i^4 - i^6\), we will follow these steps:

### Step 1: Calculate \(i^4\)
We know that:
\[ i^2 = -1 \]
Therefore:
\[ i^4 = (i^2)^2 = (-1)^2 = 1 \]

### Step 2: Substitute \(i^4\) into the expression
Now we substitute \(i^4\) back into the original expression:
\[ 2i^4 - i^6 = 2(1) - i^6 \]
This simplifies to:
\[ 2 - i^6 \]

### Step 3: Calculate \(i^6\)
We can express \(i^6\) as:
\[ i^6 = i^{4+2} = i^4 \cdot i^2 \]
From Step 1, we know \(i^4 = 1\) and \(i^2 = -1\). Therefore:
\[ i^6 = 1 \cdot (-1) = -1 \]

### Step 4: Substitute \(i^6\) back into the expression
Now we substitute \(i^6\) back into the expression:
\[ 2 - i^6 = 2 - (-1) \]
This simplifies to:
\[ 2 + 1 = 3 \]

### Final Answer
Thus, the value of \(2i^4 - i^6\) is:
\[ \boxed{3} \]

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