Answer
$P = P_{0}(1-\frac{(n-1)\rho_{0}gz}{nP_{0}})^{\frac{n}{n-1}}$
Work Step by Step
Re-arranging the given equation:
$P = C\rho^{n}$ --> $\rho = (\frac{P}{C})^{\frac{1}{n}}$
And replacing the constant with the values from the point $z = 0$:
$C = \frac{P_{0}}{\rho_{0}^{n}}$
$\rho = \rho_{0} (\frac{P}{P_0})^{\frac{1}{n}}$
And plugging it into the differential equation:
$dP = - \rho g dz$
$dP = - \rho_{0} (\frac{P}{P_0})^{\frac{1}{n}} g dz$
Integrating from $z=0, P = P_{0}$ to an arbitraty point:
$\int_{P_{0}}^{P} (\frac{P}{P_0})^{\frac{-1}{n}} dP = -\rho_{0} g \int_{0}^{z}dz$
Note: n must be different from zero, and if n = 1:
$P = P_{0} \exp(-\rho_{0}gz) $
Else if $n \ne 1$
$(\frac{P_{0}n}{n-1} \times (\frac{P}{P_{0}})^{\frac{n-1}{n}} )|_{P_{0}}^{P} = -\rho_{0}gz $
$\frac{P}{P_{0}}^{\frac{n-1}{n}} - 1 = \frac{-(n-1)\rho_{0}gz}{nP_{0}} $
$P = P_{0}(1-\frac{(n-1)\rho_{0}gz}{nP_{0}})^{\frac{n}{n-1}}$
