Answer
(a) $\sqrt{\frac{F~L}{m}}$ has units of $m/s$ which are units of speed.
(b) $v = C~\sqrt{\frac{F~L}{m}}$, where $C$ is a dimensionless constant
Work Step by Step
$F$ has units of $N$ which can also be expressed as $kg~m~s^{-2}$
$L$ has units of $m$
$m$ has units of $kg$
We can find the units of $\sqrt{\frac{F~L}{m}}$:
$\sqrt{\frac{(kg~m~s^{-2})(m)}{kg}} = \sqrt{m^2~s^{-2}} = m/s$
$\sqrt{\frac{F~L}{m}}$ has units of $m/s$ which are units of speed.
(b) Let's assume that $~C~F^a~L^b~m^c = v~$, where $C$ is a dimensionless constant.
Then: $(kg~m~s^{-2})^a~m^b~kg^c = m~s^{-1}$
We can consider the units of seconds:
$(s^{-2})^a = s^{-1}$
$-2a = -1$
$a = \frac{1}{2}$
We can consider the units of $m$:
$m^a~m^b = m^1$
$a+b = 1$
$b = 1-a$
$b = 1-\frac{1}{2}$
$b = \frac{1}{2}$
We can consider units of $kg$:
$kg^a~kg^c = kg^0$
$a+c = 0$
$c = -a$
$c = -\frac{1}{2}$
We can write the equation with the exponents:
$v = C~F^a~L^b~m^c$
$v = C~F^{1/2}~L^{1/2}~m^{-1/2}$
$v = C~\sqrt{\frac{F~L}{m}}$
