The vibration frequency of a string depends on the length L between...

Let the frequency $f$ be given by $f = k L^a F^b m^c$.

Writing the dimensions of each physical quantity:
$[f] = [T^{-1}]$
$[L] = [L]$
$[F] = [M L T^{-2}]$
$[m] = [M L^{-1}]$

Substituting the dimensions into the assumed relation:
$[T^{-1}] = [L]^a [M L T^{-2}]^b [M L^{-1}]^c$
$[T^{-1}] = [M]^{b+c} [L]^{a+b-c} [T]^{-2b}$

Equating the powers of $M$, $L$, and $T$ on both sides:
$b + c = 0$
$a + b - c = 0$
$-2b = -1$

Solving these equations:
$b = \dfrac{1}{2}$
$c = -b = -\dfrac{1}{2}$
$a = c - b = -\dfrac{1}{2} - \dfrac{1}{2} = -1$

Substituting the values of $a$, $b$, and $c$ back into the expression:
$f = k L^{-1} F^{1/2} m^{-1/2} = \dfrac{k}{L} \sqrt{\dfrac{F}{m}}$