For all `x > 0`, which of the following expressions is equivalent to `i/(sqrt(x) - i)`, where `i = sqrt(-1)`?

Options

A. `i`
B. `(sqrt(x))/(x)`
C. `(sqrt(x) - 1)/(x+ 1)`
D. `(isqrt(x) - 1)/(x + 1)`

Answer

To solve the expression \( \frac{i}{\sqrt{x} - i} \) for all \( x > 0 \), where \( i = \sqrt{-1} \), we will rationalize the denominator. Here are the steps:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{i}{\sqrt{x} - i} \]

### Step 2: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \( \sqrt{x} + i \):
\[ \frac{i}{\sqrt{x} - i} \cdot \frac{\sqrt{x} + i}{\sqrt{x} + i} \]

### Step 3: Multiply the numerators and denominators
Now we perform the multiplication:
- **Numerator**:
\[ i(\sqrt{x} + i) = i\sqrt{x} + i^2 \]
Since \( i^2 = -1 \), this simplifies to:
\[ i\sqrt{x} - 1 \]

- **Denominator**:
\[ (\sqrt{x} - i)(\sqrt{x} + i) = \sqrt{x}^2 - i^2 = x - (-1) = x + 1 \]

### Step 4: Combine the results
Now we combine the results from the numerator and denominator:
\[ \frac{i\sqrt{x} - 1}{x + 1} \]

### Step 5: Final expression
Thus, the expression \( \frac{i}{\sqrt{x} - i} \) simplifies to:
\[ \frac{i\sqrt{x} - 1}{x + 1} \]

### Conclusion
The equivalent expression for \( \frac{i}{\sqrt{x} - i} \) is:
\[ \frac{i\sqrt{x} - 1}{x + 1} \]