The Planck Mass

Let the Planck mass be:

$\begin{aligned} m_p & = G^\alpha c^\beta \hbar^\gamma \\ \left[ m_p \right] & = \left[G\right]^\alpha \left[c\right]^\beta \left[\hbar\right]^\gamma \\ M & = \left[\frac {L^3}{MT^2} \right]^\alpha \left[\frac LT \right]^\beta \left[\frac {ML^2}T \right]^\gamma \\ M & = M^{-\alpha + \gamma} L^{3\alpha + \beta + 2\gamma} T^{-2\alpha - \beta - \gamma} \end{aligned}$

$\implies \begin{cases} -\alpha + \gamma & = 1 \\ 3\alpha + \beta + 2\gamma & = 0 \\ -2\alpha - \beta - \gamma & = 0 \end{cases}$

$\implies \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} = \begin{pmatrix} -1 & 0 & 1 \\ 3 & 1 & 2 \\ -2 & -1 & -1 \end{pmatrix}^{-1} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \dfrac 12 \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}$

$\implies m_p = \boxed{\sqrt{\dfrac {c \hbar}G}}$

Verifying that the answer above does indeed have units of mass: $[\hbar]^{1/2}[c]^{1/2}[G]^{-1/2} = M^{1/2} L T^{-1/2} L^{1/2} T^{-1/2} M^{1/2} T L^{-3/2} = M.$

This can be solved for directly using the method on the String Theory wiki.