Answer
$$t_{P}=5.38 \times 10^{-44} \mathrm{s}$$
Work Step by Step
to get the solution we must find a combination of $c, G,$ and $\hbar$ that has the dimensions of time.
The dimensions of $c$ are $\left[\frac{L}{T}\right],$
the dimensions of $G$ are $\left[\frac{L^{3}}{M T^{2}}\right],$
and the dimensions of $\hbar$ are $\left[\frac{M L^{2}}{T}\right]$ . now we can use dimensional analysis.
$$t_{P}=c^{\alpha} G^{\beta} \hbar^{\gamma} \rightarrow[T]=\left[\frac{L}{T}\right]^{\alpha}\left[\frac{L^{3}}{M T^{2}}\right]^{\beta}\left[\frac{M L^{2}}{T}\right]^{\gamma}$$
$$=[L]^{\alpha+3 \beta+2 \gamma}[M]^{\gamma-\beta}[T]^{-\alpha-2 \beta-\gamma}$$
$$\alpha+3 \beta+2 \gamma=0 ; \quad \gamma-\beta=0 ; \quad-\alpha-2 \beta-\gamma=1 \quad \rightarrow $$
$$\quad \alpha+5 \beta=0 ; \quad \alpha=-1-3 \beta \rightarrow$$
$$-5 \beta=-1-3 \beta \quad \rightarrow \quad \beta=\frac{1}{2} ; \quad \gamma=\frac{1}{2} ; \quad \alpha=-\frac{5}{2}$$
$$t_{P}=c^{-5 / 2} G^{1 / 2} \hbar^{1 / 2}=\sqrt{\frac{G \hbar}{c^{5}}}=$$
$$=\sqrt{\frac{\left(6.67 \times 10^{-11} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}\right) \frac{1}{2 \pi}\left(6.63 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}\right)}{\left(3.00 \times 10^{8} \mathrm{m} / \mathrm{s}\right)^{5}}}$$
$$5.38 \times 10^{-44} \mathrm{s}$$
