Hi thetada!
For most questions, like the new pressure, you do need γ. Either you give algebraic answers, or you must suppose some γ as you did, since the questions give numerical values. Monoatomic, or diatomic like air, or anything else.
PVγ=const is convenient between P and V. Other formulas are more direct for other variables, they hold for adiabatic transformations of perfect gases and result from the combination with PV/T=const:
P=TCp/R×const
V=TCv/R×const
where Cv/R is 3/2 for a monoatomic gas (3 translations) and 5/2 for a diatomic one without vibrations (adds 2 rotations), and Cp/R = 1 + Cv/R, which is the same as H=U+PV.
To remember it: Cp gives P and Cv gives V.
The change of internal energy is ΔU=ΔT×Cv
and of enthalpy ΔH=ΔT×Cp which gives you the work.
Most often, the temperature ratio is best computed first, and everything else deduced from it, first as ratios.