Express in the form of complex number `i^9+i^(19)`

Answer

To express \( i^9 + i^{19} \) in the form of a complex number, we can follow these steps:

### Step 1: Understand the powers of \( i \)
The powers of \( i \) (the imaginary unit) cycle every four terms:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \) (and the cycle repeats)

### Step 2: Simplify \( i^9 \)
To simplify \( i^9 \), we can find the remainder when 9 is divided by 4:
\[ 9 \div 4 = 2 \quad \text{(remainder 1)} \]
Thus, \( i^9 = i^{4 \cdot 2 + 1} = (i^4)^2 \cdot i^1 = 1^2 \cdot i = i \).

### Step 3: Simplify \( i^{19} \)
Similarly, for \( i^{19} \):
\[ 19 \div 4 = 4 \quad \text{(remainder 3)} \]
Thus, \( i^{19} = i^{4 \cdot 4 + 3} = (i^4)^4 \cdot i^3 = 1^4 \cdot (-i) = -i \).

### Step 4: Combine the results
Now we can combine the results from steps 2 and 3:
\[ i^9 + i^{19} = i + (-i) = i - i = 0. \]

### Step 5: Express in standard form
The standard form of a complex number is \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. In this case:
\[ 0 + 0i. \]

### Final Result
Thus, the expression \( i^9 + i^{19} \) in the form of a complex number is:
\[ \boxed{0 + 0i}. \]