Simplify:
`i+ 2i^(2) + 3i^(3) + i^(4)`

Answer

To simplify the expression \( i + 2i^2 + 3i^3 + i^4 \), we will follow these steps:

### Step 1: Identify the powers of \( i \)
We know the following properties of the imaginary unit \( i \):
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

### Step 2: Substitute the powers of \( i \) into the expression
Using the properties identified, we can substitute the powers of \( i \) in the expression:
\[ i + 2i^2 + 3i^3 + i^4 = i + 2(-1) + 3(-i) + 1 \]

### Step 3: Simplify the expression
Now, we simplify each term:
\[ = i - 2 - 3i + 1 \]

### Step 4: Combine like terms
Next, we combine the real parts and the imaginary parts:
- Real parts: \( -2 + 1 = -1 \)
- Imaginary parts: \( i - 3i = -2i \)

Putting it all together, we have:
\[ -1 - 2i \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{-1 - 2i} \]