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.

I am going to set the mantissa of this code to a MILLION digits. It’s basically 5, followed by a decimal point, followed by a million digits after that. Every method I tried to compute the first million digits of my number results in an overflow error. I don’t get the same error when computing the last million digits. To me the first and last million digits is the same size of the number and they both shouldn’t overflow. That “100” you wrote in your code should be at least 50, 000. I copied and pasted your code into my notebook and set the number to 10, 000. And I get an error “precision depth exceeded.”

I haven't found a way to get the first digits of my number and it's so EXTREMELY HARD that I must as well forget about them and focus on the last digits. I typed my expression in and it gave me something mod^2 power, 1977326743 followed by a million zeros and these ARE NOT THE LAST MILLION DECIMAL DIGITS I AM EXPECTING!

I calculate the last digits by using mod 10^50000 something. How do I calculate the FIRST 50, 000 digits of my expression (96717311574016^16777216^4)^8^4)^7)^2 * 7^11

my notebook is 193 KB, and I bought it directly from Mathematica. I dont remember saving it into this computer. I found on the introductory five minutes.nb then I clicked on it and it went into my notes. I made it because when I type in my expression in, it gives me "cell contents are too large for direct browser rendering. Try rendering your browser anyway or download this notebook for full functionality." I click on it in my notes but I can't open it, I just get duplicates of it. How can I type my expression in when I can't even open it?!?

the notebook five minutes.nb is in my notes but i cant open it; clicking it just duplicates it in my notes

it gives me syntax error: ( can't be followed by mod (power mod(96717311574016, 16777216^4 * 8^4 * 7 * 2, 10^(10^6)) * 7^11, 10^(10^6))