Answer
The dimensions of $l_p$ are [L].
$l_p \approx 10^{-35} ~m$
Work Step by Step
The dimensions of $l_p$ are $\sqrt{\frac{[L]^3\times[M]\times[L]^2\times[T]^3}{[M]\times[T]^2\times[T]\times[L]^3}}=\sqrt{[L]^2} = [L]$
$l_p = \sqrt{\frac{Gh}{c^3}} = \sqrt{\frac{(6.67\times10^{-11})(6.63\times10^{-34})}{(3.00\times10^8)^3}} = \sqrt{1.64\times10^{-69}}= 4.05\times10^{-35} ~m \approx 10^{-35}~m$
