Which of the following is equivalent to `2i^(2)+3i^(3)`?

Options

A. `-2-3i`
B. `2-3i`
C. `-2+3i`
D. `2+3i`

Answer

To solve the expression \(2i^2 + 3i^3\), we will follow these steps:

### Step 1: Evaluate \(i^2\)
We know that \(i\) is defined as the square root of \(-1\). Therefore:
\[ i^2 = -1 \]

### Step 2: Evaluate \(i^3\)
We can express \(i^3\) in terms of \(i^2\):
\[ i^3 = i^2 \cdot i = (-1) \cdot i = -i \]

### Step 3: Substitute \(i^2\) and \(i^3\) into the expression
Now we can substitute \(i^2\) and \(i^3\) back into the original expression:
\[ 2i^2 + 3i^3 = 2(-1) + 3(-i) \]

### Step 4: Simplify the expression
Calculating each term, we get:
\[ 2(-1) = -2 \]
\[ 3(-i) = -3i \]
Thus, combining these results:
\[ 2i^2 + 3i^3 = -2 - 3i \]

### Conclusion
The expression \(2i^2 + 3i^3\) simplifies to:
\[ -2 - 3i \]

### Final Answer
The equivalent expression is \(-2 - 3i\).

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