Let `x-1/x=(sqrt2)i` where `i=sqrt(-1)` . Then the value of `x^(2187)-1/x^(2187)` is :

Options

A. `isqrt2`
B. `-isqrt2`
C. `-2`
D. `i/sqrt2`

Answer

To solve the problem \( x - \frac{1}{x} = \sqrt{2}i \), where \( i = \sqrt{-1} \), and find the value of \( x^{2187} - \frac{1}{x^{2187}} \), we will follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x - \frac{1}{x} = \sqrt{2}i \]

2. **Square both sides:**
\[ \left(x - \frac{1}{x}\right)^2 = (\sqrt{2}i)^2 \]
Expanding the left side:
\[ x^2 - 2 + \frac{1}{x^2} = -2 \]
Thus, we have:
\[ x^2 + \frac{1}{x^2} - 2 = -2 \]
Simplifying gives:
\[ x^2 + \frac{1}{x^2} = 0 \]

3. **Multiply through by \( x^2 \) to eliminate the fraction:**
\[ x^4 + 1 = 0 \]
This implies:
\[ x^4 = -1 \]

4. **Find the roots of \( x^4 = -1 \):**
The solutions to this equation are:
\[ x = e^{i\frac{\pi}{4}}, e^{i\frac{3\pi}{4}}, e^{i\frac{5\pi}{4}}, e^{i\frac{7\pi}{4}} \]

5. **Now, we need to find \( x^{2187} - \frac{1}{x^{2187}} \):**
First, we calculate \( x^{2187} \):
Since \( x^4 = -1 \), we can express \( x^{2187} \) in terms of \( x^4 \):
\[ 2187 \mod 4 = 3 \]
Thus:
\[ x^{2187} = x^3 \]

6. **Next, calculate \( \frac{1}{x^{2187}} \):**
Since \( x^{2187} = x^3 \):
\[ \frac{1}{x^{2187}} = \frac{1}{x^3} \]

7. **Now, find \( x^3 - \frac{1}{x^3} \):**
Using the identity:
\[ x^3 - \frac{1}{x^3} = \left(x - \frac{1}{x}\right)\left(x^2 + 1 + \frac{1}{x^2}\right) \]
We already found that \( x - \frac{1}{x} = \sqrt{2}i \) and \( x^2 + \frac{1}{x^2} = 0 \), thus:
\[ x^2 + 1 + \frac{1}{x^2} = 1 \]
Therefore:
\[ x^3 - \frac{1}{x^3} = \sqrt{2}i \cdot 1 = \sqrt{2}i \]

8. **Final result:**
Thus, the value of \( x^{2187} - \frac{1}{x^{2187}} \) is:
\[ \sqrt{2}i \]