Distance from earth to moon is about 384,000,000 meters. Speed of light is about 300,000,000 m/s, so at the speed of light it would take 384/300 = 1.28 seconds. This is confirmed by radar, laser and radio communication.

However thanks to relativistic time dilation the clock on the rocket would measure a time

t' = t/√(1-v^{2}/c^{2}) compared with a stationary clock so if v = c/√2 then v^{2}/c^{2} = ½ and t'= 1.28/1.41 ≈ 0.9 seconds.