Answer
$v=1.16\times 10^{-8}\;\rm m/s=41.67\;\mu m/h$
Work Step by Step
Let's assume that your hair grows 3 cm per month. Now we need to find the growth speed in meters per second and in micrometers per hour.
$$v=\rm \dfrac{3\;cm}{1\;month} \cdot \left(\dfrac{1\;m}{100\;cm}\right)\cdot \left(\dfrac{1\;month}{30\;day}\right)\cdot \left(\dfrac{1\;day}{24\;h}\right)\cdot \left(\dfrac{1\;h}{60^2\;s}\right)$$
The unit of $\rm cm,\;month,\;day,\;h$ cancels and the units of $\rm m/s$ remain.
Thus,
$$\boxed{v=1.16\times10^{-8}\;\rm m/s }$$
Now we need to find the speed in micrometers per hour.
$$v=\rm \dfrac{1.16\times10^{-8}\;m}{1\;s}\;\cdot\left(\dfrac{10^6\;\mu m}{1\;m}\right)\cdot \left(\dfrac{60^2\;s}{1\;h}\right) $$
The unit of $\rm m,{\rm and}\;s$ cancel, and the units of $\rm \mu m/h$ remain.
Thus,
$$\boxed{v=41.67\;\rm \mu m/h}$$
