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A physical quantity of the dimensions of length that can be formed out of c, G and $\frac{e^2}{4\pi\varepsilon_0}$ is [c is velocity of light, G is universal constant of gravitation and e is charge]
Explanation
Match powers of M, L and T to get $x = -2$, $y = z = 1/2$.
Detailed Solution
Let $\frac{e^2}{4\pi\varepsilon_0} = A = [ML^3T^{-2}]$
$l = c^xG^yA^z$
$L = [LT^{-1}]^x[M^{-1}L^3T^{-2}]^y[ML^3T^{-2}]^z$
$-y + z = 0 \Rightarrow y = z$ ...(i)
$x + 3y + 3z = 1$ ...(ii)
$-x - 2y - 2z = 0$ ...(iii)
From (i), (ii) and (iii): $z = y = \frac{1}{2}$, $x = -2$
So the length is $\frac{1}{c^2}\left[G\frac{e^2}{4\pi\varepsilon_0}\right]^{1/2}$.