= Solution
Use dimensions $\mathsf M,\mathsf L,\mathsf T$ for <mass>, <length> and <time>. The <reduced Planck constant>, <gravitational constant> and <speed of light> have dimensions
$$
[\hbar]=\mathsf M\mathsf L^2\mathsf T^{-1},\qquad
[G]=\mathsf L^3\mathsf M^{-1}\mathsf T^{-2},\qquad
[c]=\mathsf L\mathsf T^{-1}.
$$
For $l_p=\hbar^\alpha G^\beta c^\gamma$ to have the dimensions of a <length>, <dimensional analysis> requires
$$
\alpha-\beta=0,\qquad
2\alpha+3\beta+\gamma=1,\qquad
-\alpha-2\beta-\gamma=0.
$$
The first equation gives $\beta=\alpha$; the third gives $\gamma=-3\alpha$. Substitution in the second gives $2\alpha=1$. Hence
$$
\boxed{\alpha=\frac12,\qquad\beta=\frac12,\qquad\gamma=-\frac32,
\qquad l_p=\sqrt{\frac{\hbar G}{c^3}}.}
$$
With $\hbar=1.054571817\times10^{-34}\,\mathrm{J\,s}$, $G=6.67430\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}$ and $c=2.99792458\times10^8\,\mathrm{m\,s^{-1}}$, the <Planck length> is
$$
\boxed{l_p\simeq1.6163\times10^{-35}\,\mathrm m.}
$$
The proposed monomial fixes the dimensionless prefactor to one; <dimensional analysis> on its own cannot select an arbitrary dimensionless multiplier.
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