If `i=sqrt(-1)` , which of the following is equivalent to `(14)/(2-sqrt(10)i)`?

Options

A. `2+sqrt(10)i`
B. `2-(sqrt(10))/(14)i`
C. `(2-sqrt(10)i)/(14)`
D. `28-14sqrt(10)i`

Answer

To solve the expression \(\frac{14}{2 - \sqrt{10}i}\), we will multiply the numerator and the denominator by the conjugate of the denominator.

### Step-by-step Solution:

1. **Identify the conjugate of the denominator**:
The denominator is \(2 - \sqrt{10}i\). Its conjugate is \(2 + \sqrt{10}i\).

2. **Multiply the numerator and denominator by the conjugate**:
\[ \frac{14}{2 - \sqrt{10}i} \cdot \frac{2 + \sqrt{10}i}{2 + \sqrt{10}i} = \frac{14(2 + \sqrt{10}i)}{(2 - \sqrt{10}i)(2 + \sqrt{10}i)} \]

3. **Calculate the denominator using the difference of squares**:
\[ (2 - \sqrt{10}i)(2 + \sqrt{10}i) = 2^2 - (\sqrt{10}i)^2 = 4 - (\sqrt{10})^2(i^2) \]
Since \(i^2 = -1\), we have:
\[ 4 - 10(-1) = 4 + 10 = 14 \]

4. **Calculate the numerator**:
\[ 14(2 + \sqrt{10}i) = 28 + 14\sqrt{10}i \]

5. **Combine the results**:
\[ \frac{28 + 14\sqrt{10}i}{14} \]

6. **Simplify the expression**:
\[ = \frac{28}{14} + \frac{14\sqrt{10}i}{14} = 2 + \sqrt{10}i \]

Thus, the expression \(\frac{14}{2 - \sqrt{10}i}\) simplifies to \(2 + \sqrt{10}i\).

### Final Answer:
\[ 2 + \sqrt{10}i \]