A monoatomic gas at a pressure P having a volume V  expands isothermally to a volume 2V  and then adiabatically to a volume 16 V. The final pressure of the gas is (Take $\gamma$ = 5/3)

For the isothemal expansion, we know that
PV = constant

$\Rightarrow {\mathrm{P}}_1{\mathrm{V}}_1={\mathrm{P}}_2{\mathrm{V}}_2$

$\Rightarrow \mathrm{PV}={\mathrm{P}}_2\left(2\mathrm{V}\right)$

$\Rightarrow {\mathrm{P}}_2=\frac{\mathrm{P}}{2}$

For further adiabatic expansion to 16 V,

${\mathrm{PV}}^{\gamma }=\text{constant}$

(for an adiabatic process) and given, $\gamma =\frac{5}{3}$

$\Rightarrow P_1{V}_1^{\gamma }=P_2{V}_2^{\gamma }$

$\Rightarrow \left(\frac{\mathrm{P}}{2}\right)(2\mathrm{V}{)}^{\frac{5}{3}}={\mathrm{P}}_2(16\mathrm{V}{)}^{\frac{5}{3}}$

$\Rightarrow P_2=\left(\frac{P}{2}\right){\left(\frac{1}{8}\right)}^{\frac{5}{3}}$

$=\frac{P}{2^6}=\frac{P}{64}$