transmission line - Derivation of Characteristic Impedance?

I start from the telegrapher's equation: \$-\frac{dV(z)}{dz}=(R'+j\omega L')I(z)\$, where \$V(z)\$ and \$I(z)\$ are the phasors of voltage and current respectively, in the transmission line model. \$R'\$ and \$L'\$ are resistance per unit length and inductance per unit length respectively.

The solution to the wave equation \$\frac{d^2V(z)}{dz^2}-\gamma^2V(z)=0\$ where \$\gamma=\sqrt{(R'+j\omega L')(G'+j\omega C')} \$ has the form \$V(z)=V_o^+e^{-\gamma z}+V_o^-e^{\gamma z} \$. \$G'\$ and \$C'\$ are respectively the conductance per unit length and capacitance per unit length of the transmission line.

From the telegrapher's equation we get: \$-\frac{dV(z)}{dz}=\gamma V_o^+e^{-\gamma z}-\gamma V_o^-e^{\gamma z}=\gamma (V_o^+e^{-\gamma z}-V_o^-e^{\gamma z})=(R'+j\omega L')I(z)\$

\$I(z)=\frac{\gamma}{R'+j\omega L'}(V_o^+e^{-\gamma z}-V_o^-e^{\gamma z})\$

...and I'm stuck here.

Given that characteristic impedance \$\frac{V_o^+}{I_o^+}=Z_o=\frac{V_o^-}{I_o^-}\$, how do I arrive at \$Z_o=\frac{R'+j\omega L'}{\gamma}=\sqrt{\frac{R'+j\omega L'}{G'+j\omega C'}}\$?

I'm not sure how to get \$Z_o\$ from \$\frac{V_o^+e^{-\gamma z}+V_o^-e^{\gamma z}}{I(z)}=\frac{V_o^+e^{-\gamma z}-V_o^-e^{\gamma z}}{I_o^+e^{-\gamma z}+I_o^-e^{\gamma z}}\$