Evaluate ` 2i^(2)+ 6i^(3)+3i^(16) -6i^(19) + 4i^(25) `

Answer

To evaluate the expression \( 2i^2 + 6i^3 + 3i^{16} - 6i^{19} + 4i^{25} \), we will use the properties of the imaginary unit \( i \), where \( i = \sqrt{-1} \) and the powers of \( i \) cycle every four terms:

- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \) (and so on...)

### Step-by-step Solution:

1. **Evaluate \( i^2 \)**:
\[ i^2 = -1 \]
Substitute into the expression:
\[ 2i^2 = 2(-1) = -2 \]

2. **Evaluate \( i^3 \)**:
\[ i^3 = -i \]
Substitute into the expression:
\[ 6i^3 = 6(-i) = -6i \]

3. **Evaluate \( i^{16} \)**:
Since \( 16 \) is a multiple of \( 4 \):
\[ i^{16} = (i^4)^4 = 1^4 = 1 \]
Substitute into the expression:
\[ 3i^{16} = 3(1) = 3 \]

4. **Evaluate \( i^{19} \)**:
\( 19 \mod 4 = 3 \), so:
\[ i^{19} = i^3 = -i \]
Substitute into the expression:
\[ -6i^{19} = -6(-i) = 6i \]

5. **Evaluate \( i^{25} \)**:
\( 25 \mod 4 = 1 \), so:
\[ i^{25} = i^1 = i \]
Substitute into the expression:
\[ 4i^{25} = 4(i) = 4i \]

6. **Combine all the terms**:
Now, we combine all the evaluated terms:
\[ -2 - 6i + 3 + 6i + 4i \]
Combine the real parts:
\[ -2 + 3 = 1 \]
Combine the imaginary parts:
\[ -6i + 6i + 4i = 4i \]
Therefore, the final result is:
\[ 1 + 4i \]

### Final Answer:
\[ 1 + 4i \]