Find the multiplicative inverse of 5 + 12i, and check that the product of the number and its…

$\frac{1}{a + ib} = \frac{a - ib}{a^2 + b^2}$. To divide, multiply the top and the bottom by the bottom with the sign of $i$ changed. The powers of $i$ repeat in fours: $i, -1, -i, 1$. So $i^n = i^r$, where $r$ is the remainder of $n$ on division by $4$.