Question for the experts:

How does the cyclicity work?

I wonder if the cyclicity of the "powered up" numbers goes well beyond units or tens digits. Actually once I've computed one power can I reuse it? Illustration [inspired by

math revolution post]:

The thousands digits of 7^k cycles through......?−−>?−−>?−−−>2−−−>?−−>…

The hundreds digits of 7^k cycles through.......?−−>?−−>3−−−>4−−−>?−−>…

The tens digit of 7^k cycles through.............0−−>4−−>4−−−>0−−−>0−−>…

The units digit of 7^k cycles through............7−−>9−−>3−−−>1−−−>7−−>…

The only calculation I've made are 7^3 and 7^4 and with that I've partly filled the 7 cycle up to the thousands! But then I've checked for 2401^2 but it doesn't work...

Help please... Thx

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