If the multiplicative inverse of a complex number is ` (sqrt2 +5i)/17` ,then the complex number is

Options

A. `(sqrt2 -5i)/17`
B. `(sqrt2 +5i)/29`
C. `17/27 (sqrt2 -5i)`
D. `17/27 (sqrt2 +5i)`

Answer

To find the complex number \( z \) given its multiplicative inverse \( \frac{\sqrt{2} + 5i}{17} \), we can follow these steps:

### Step 1: Understand the relationship between a complex number and its multiplicative inverse
The multiplicative inverse of a complex number \( z \) is given by:
\[ \frac{1}{z} = \frac{\sqrt{2} + 5i}{17} \]
From this relationship, we can express \( z \) as:
\[ z = \frac{17}{\sqrt{2} + 5i} \]

### Step 2: Rationalize the denominator
To eliminate the imaginary unit \( i \) from the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \( \sqrt{2} - 5i \):
\[ z = \frac{17(\sqrt{2} - 5i)}{(\sqrt{2} + 5i)(\sqrt{2} - 5i)} \]

### Step 3: Simplify the denominator
Using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \):
\[ (\sqrt{2})^2 - (5i)^2 = 2 - 25(-1) = 2 + 25 = 27 \]

### Step 4: Substitute back into the expression for \( z \)
Now substituting back, we have:
\[ z = \frac{17(\sqrt{2} - 5i)}{27} \]

### Step 5: Write the final expression for \( z \)
Thus, the complex number \( z \) can be expressed as:
\[ z = \frac{17\sqrt{2}}{27} - \frac{85i}{27} \]

### Final Answer
The complex number \( z \) is:
\[ z = \frac{17\sqrt{2}}{27} - \frac{85i}{27} \]