Simplify `i^18-3i^7 +i^2 (1 +i^4) (-i)^26`.

Answer

To simplify the expression \( i^{18} - 3i^{7} + i^{2}(1 + i^{4})(-i)^{26} \), we can follow these steps:

### Step 1: Simplify \( i^{18} \)
We know that \( i^4 = 1 \). Therefore, we can express \( i^{18} \) as:
\[ i^{18} = i^{16} \cdot i^{2} = (i^{4})^{4} \cdot i^{2} = 1^{4} \cdot i^{2} = i^{2} = -1 \]

### Step 2: Simplify \( -3i^{7} \)
We can express \( i^{7} \) as:
\[ i^{7} = i^{4} \cdot i^{3} = 1 \cdot i^{3} = i^{3} \]
Now, \( i^{3} = -i \), so:
\[ -3i^{7} = -3(-i) = 3i \]

### Step 3: Simplify \( i^{2}(1 + i^{4})(-i)^{26} \)
First, simplify \( i^{4} \):
\[ i^{4} = 1 \]
So, \( 1 + i^{4} = 1 + 1 = 2 \).

Next, simplify \( (-i)^{26} \):
\[ (-i)^{26} = (-1)^{26} \cdot i^{26} = 1 \cdot i^{26} = i^{26} \]
Now, simplify \( i^{26} \):
\[ i^{26} = i^{24} \cdot i^{2} = (i^{4})^{6} \cdot i^{2} = 1^{6} \cdot i^{2} = i^{2} = -1 \]
Thus, \( (-i)^{26} = -1 \).

Putting it all together:
\[ i^{2}(1 + i^{4})(-i)^{26} = (-1)(2)(-1) = 2 \]

### Step 4: Combine all parts
Now we can substitute back into the original expression:
\[ i^{18} - 3i^{7} + i^{2}(1 + i^{4})(-i)^{26} = -1 + 3i + 2 \]
Combine the real parts:
\[ (-1 + 2) + 3i = 1 + 3i \]

### Final Answer
The simplified expression is:
\[ \boxed{1 + 3i} \]