The pressure and density of a diatomic gas ($\gamma = \frac{7}{5}$) changes adiabatically from $(P, \rho)$ to $(P', \rho')$. If $\frac{\rho'}{\rho} = 32$ then $\frac{P'}{P}$ should be
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Use the power of 2 for quick calculation: $32 = 2^5$, so $32^{7/5} = 2^7 = 128$.