I think that

(((96717311574016^16777216^4)^8^4)^7)^2==96717311574016^(16777216^4*8^4*7*2)

You must check that and convince yourself whether that is correct or not. As you have seen, Mathematica cannot simply directly calculate both those values and compare them for equality.

If that is correct then

Mod[PowerMod[96717311574016,16777216^4*8^4*7*2,10^(10^4)]*7^11,10^(10^4)]

returns the last 10 thousand digits of the result in a few seconds.

Study all the details of the documentation for PowerMod carefully and then find at least three completely different ways that you can try to check whether I have made any mistake or if the result might be incorrect.

Changing from 10 thousand digits to 1 million digits required changing 10^(10^4) to 10^(10^6) in two places. The output is then large enough that MMA doesn't immediately show you all of the result on the screen so adding //InputForm may overcome that. That may pop up a little window asking you if you really want an output that large. If so then you confirm you want that.

Try

Mod[PowerMod[96717311574016,16777216^4*8^4*7*2,10^(10^6)]*7^11,10^(10^6)]//InputForm

and see if that works for you.