The correct order of nuclear decay modes for the change from U-238 to U-234 is:

1. Alpha Decay: U-238 undergoes alpha decay, emitting an alpha particle (consisting of two protons and two neutrons) and transforming into Th-234.

2. Beta Decay: Th-234 is unstable and undergoes beta decay, emitting a beta particle (an electron) and transforming into Pa-234.

3. Beta Decay: Pa-234 is also unstable and undergoes beta decay, emitting a beta particle and transforming into U-234.

Therefore, the complete decay chain is:

U-238 → α-decay → Th-234 → β-decay → Pa-234 → β-decay → U-234