# How do you find the multiplicative inverse of 5+2i in standard form?

Jan 13, 2016

The explanation is given below.

#### Explanation:

The multiplicative inverse of $z$ is $\frac{1}{z}$ where $z \left(\frac{1}{z}\right) = 1$

In our problem, we have $z = 5 + 2 i$ we need to find $\frac{1}{z}$

Which would be, $\frac{1}{5 + 2 i}$

$\frac{1}{5 + 2 i} = \frac{1}{5 + 2 i} \cdot \frac{5 - 2 i}{5 - 2 i}$

Multiply numerator and denominator by the conjugate of the denominator and make the denominator a real number.

$\frac{1}{5 + 2 i} = \frac{5 - 2 i}{{5}^{2} - {\left(2 i\right)}^{2}}$

$\frac{1}{5 + 2 i} = \frac{5 - 2 i}{25 + 4}$

$\frac{1}{5 + 2 i} = \frac{5 - 2 i}{29}$

$\frac{1}{5 + 2 i} = \frac{5}{29} - \frac{2}{29} i$

The multiplicative inverse is $\frac{5}{29} - \frac{2}{29} i$#