Question

Find the multiplicative inverse of \( \sqrt{5} + 3i \)

Hint:

To find the multiplicative inverse of a complex number, multiply the numerator and denominator by its conjugate and simplify.

Answer 1

Step 1: Write the inverse of the complex number.
The multiplicative inverse of a complex number \( z = a + bi \) is given by: \[ z^{-1} = \frac{1}{z} = \frac{1}{a + bi} \] For \( z = \sqrt{5} + 3i \), the multiplicative inverse will be: \[ z^{-1} = \frac{1}{\sqrt{5} + 3i} \]
Step 2: Multiply by the conjugate.
To simplify, we multiply both the numerator and denominator by the conjugate of \( \sqrt{5} + 3i \), which is \( \sqrt{5} - 3i \): \[ z^{-1} = \frac{1}{\sqrt{5} + 3i} \times \frac{\sqrt{5} - 3i}{\sqrt{5} - 3i} \] \[ z^{-1} = \frac{\sqrt{5} - 3i}{(\sqrt{5})^2 + (3i)^2} = \frac{\sqrt{5} - 3i}{5 - 9} = \frac{\sqrt{5} - 3i}{-4} \]
Step 3: Simplify the result.
Now, simplify the expression: \[ z^{-1} = -\frac{\sqrt{5}}{4} + \frac{3i}{4} \]
Step 4: Conclusion.
The multiplicative inverse of \( \sqrt{5} + 3i \) is: \[ z^{-1} = -\frac{\sqrt{5}}{4} + \frac{3i}{4} \]