Find the multiplicative inverse of z = √5 + 3i in the form a + ib.

For $z = a + ib$: the conjugate is $\bar z = a - ib$ and the modulus is $|z| = \sqrt{a^2 + b^2}$. Then $z\bar z = |z|^2$, $|z_1 z_2| = |z_1||z_2|$, and $\overline{z_1 z_2} = \bar z_1 \bar z_2$. To divide, multiply the top and the bottom by the conjugate of the bottom: $\dfrac{z_1}{z_2} = \dfrac{z_1 \bar z_2}{|z_2|^2}$.