What is the value of `(-sqrt(-1))^(4n+3)+(i^(41)+1^(-257))^(9),` when n inN` ?

Options

A. 0
B. 1
C. It is parallel to the y-axis
D. `-i`

Explanation

To solve the expression \((- \sqrt{-1})^{4n+3} + (i^{41} + 1^{-257})^9\), we can break it down into manageable parts.

### Step-by-Step Solution:

1. **Simplify \(-\sqrt{-1}\)**:
\[ -\sqrt{-1} = -i \]
Therefore, we rewrite the first part:
\[ (-\sqrt{-1})^{4n+3} = (-i)^{4n+3} \]

2. **Use properties of exponents**:
The expression \((-i)^{4n+3}\) can be simplified using the fact that \((-i) = e^{-i\frac{\pi}{2}}\):
\[ (-i)^{4n+3} = (e^{-i\frac{\pi}{2}})^{4n+3} = e^{-i\frac{\pi}{2}(4n+3)} = e^{-i(2n\pi + \frac{3\pi}{2})} = e^{-i\frac{3\pi}{2}} = i \]

3. **Simplify \(i^{41}\)**:
To simplify \(i^{41}\), we can use the periodicity of \(i\) (where \(i^4 = 1\)):
\[ i^{41} = i^{4 \cdot 10 + 1} = (i^4)^{10} \cdot i^1 = 1^{10} \cdot i = i \]

4. **Simplify \(1^{-257}\)**:
Since any non-zero number raised to a negative exponent is equal to its reciprocal:
\[ 1^{-257} = \frac{1}{1^{257}} = 1 \]

5. **Combine \(i^{41}\) and \(1^{-257}\)**:
Now we can combine the results:
\[ i^{41} + 1^{-257} = i + 1 \]

6. **Raise to the power of 9**:
Now we need to find \((i + 1)^9\). First, we can express \(i + 1\) in polar form:
\[ i + 1 = \sqrt{2} \left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) \]
Therefore:
\[ (i + 1)^9 = (\sqrt{2})^9 \left(\cos\left(\frac{9\pi}{4}\right) + i\sin\left(\frac{9\pi}{4}\right)\right) \]
\[ = 2^{\frac{9}{2}} \left(\cos\left(\frac{9\pi}{4}\right) + i\sin\left(\frac{9\pi}{4}\right)\right) \]

7. **Simplify \(\frac{9\pi}{4}\)**:
Since \(\frac{9\pi}{4} = 2\pi + \frac{\pi}{4}\), we have:
\[ \cos\left(\frac{9\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}, \quad \sin\left(\frac{9\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

8. **Final calculation**:
Thus:
\[ (i + 1)^9 = 2^{\frac{9}{2}} \left(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\right) = 2^{\frac{9}{2}} \cdot \frac{1}{\sqrt{2}}(1 + i) = 2^4(1 + i) = 16(1 + i) \]

9. **Combine both parts**:
Now we combine both parts:
\[ (-i)^{4n+3} + (i + 1)^9 = i + 16(1 + i) = i + 16 + 16i = 16 + 17i \]

### Final Answer:
\[ \text{The value is } 16 + 17i. \]