10^2500 is a 2501 digit number, 9^2500 is a 2386 digit number; it is the tiniest tiniest fraction.

 

All that said Michel - it would not massively surprise me if someone has come up with a clever and more importantly quick way of calculating it

Yes I know it is not a simple question and I am also pretty sure there is a much simpler way to calculate it but 1. my skills are reduced to a minimum and 2. when I cannot solve a math question in a few minutes I tend to abandon, it must be a reminiscence of the way my exams were conducted: if you cannot answer immediately, jump to the next question otherwise you are burned.

 

How did you figure that "9^2500 is a 2386 digit number"?

Yes I know it is not a simple question and I am also pretty sure there is a much simpler way to calculate it but 1. my skills are reduced to a minimum and 2. when I cannot solve a math question in a few minutes I tend to abandon, it must be a reminiscence of the way my exams were conducted: if you cannot answer immediately, jump to the next question otherwise you are burned.

 

How did you figure that "9^2500 is a 2386 digit number"?

 

I looked it up - Wolfram Alpha for all your calculating needs and the OP stated it was that long above

 

How did you figure that "9^2500 is a 2386 digit number"?

I used the calculator on my Linux system; the Windows calc would probably do the same.

I am still working on this, but I am not going to try to do this all in one night. Finding 3^256 is ridiculous(can't imagine 3^5000). With the weekend coming up I will have time to work on this. I am trying to find other more efficient ways to do it, but as of now, doing out the problem in the traditional was has seemed to be the "Quickest" (Although I will say this process is anything but quick)

I am still working on this, but I am not going to try to do this all in one night. Finding 3^256 is ridiculous(can't imagine 3^5000). With the weekend coming up I will have time to work on this. I am trying to find other more efficient ways to do it, but as of now, doing out the problem in the traditional was has seemed to be the "Quickest" (Although I will say this process is anything but quick)

 

As I mentioned above - but you may have missed; I think this is the quickest way

((((((((((((((((3^2)^2)^2)*3)^2)*3)^2)*3)^2)^2)^2)^2)*3)^2)^2)^2)

 

There are some phenomenal size multiplications to be done - but that will always be the case; however there is no preparation calcs needed. By that I mean you do not have to work out things like 3^256 - just 16 multiplications.

 

Is that the silliest use of the word "just" in the last few years

It's just bad. Squaring as L Meow is doing may not be as few multiplications as your method imatfaal, but it seems like a good method to me, and I think the number of digits to write will not change much as Acme said in #13.

((((((((((((((((3^2)^2)^2)*3)^2)*3)^2)*3)^2)^2)^2)^2)*3)^2)^2)^2)

 

I would think this is the quickest way

 

I will leave why this is what I have come up with as an exercise - unless you really want to know in which case tell me

 

=81^1250

yeah this way is most easy

9^2500=81^1250=6561^625

(6561^5)^125= Best Of Luck Dude :')

The easiest way is probably what Lightmeow has already hit on:

 

3^5000 = 3^(4096 + 512 + 256 + 128 + 8) = 3^4096 * 3^512 * 3^256 * 3^128 * 3^8 = 3^(2^12) * 3^(2^9) * 3^(2^8) * 3^(2^7) * 3^(2^3)

 

Then each of the successive 3^(2^i) can be calculated inductively, by squaring the last result. This is how computers do powers, in general.

I'm waiting for the OP to hand in the result and get told "Oh,! sorry, I meant 5000^3. Oops!".