How to find the exact answer of 3^5000 by hand

[michel123456]

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"?

[imatfaal]

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

[EdEarl]

 

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.

[Lightmeow]

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)

[imatfaal]

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

[EdEarl]

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.

[Rajnish Kaushik]

((((((((((((((((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 :')

[uncool]

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.

[John Cuthber]

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