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

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Title says it all. I've been challenged by my teacher to do this. I am currently doing it out by hand, it it is extremely tedious and annoying. Are there other more efficient ways to do this?

(P.S. I know I can put it into wolfram, I do have the answer. I just need to figure it out by hand)

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That doesn't sound reasonable. I can think of two possibilities here:

1) You annoyed the teacher and (s)he is giving you something that will keep you busy a long time; or

2) You are supposed to find the remainder mod something. For example mod 4, every power of 3 is either 1 or 3 and you can easily figure out which ones are which.

What do you think? Anything to my theory? Otherwise, get a box of pencils and a stack of paper. Or perhaps are you supposed to just estimate its order of magnitude? That would be another more reasonable request.

Edited by wtf
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That doesn't sound reasonable. I can think of two possibilities here:

1) You annoyed the teacher and (s)he is giving you something that will keep you busy a long time; or

2) You are supposed to find the remainder mod something. For example mod 4, every power of 3 is either 1 or 3 and you can easily figure out which ones are which.

What do you think? Anything to my theory? Otherwise, get a box of pencils and a stack of paper. Or perhaps are you supposed to just estimate its order of magnitude? That would be another more reasonable request.

No, she just wants me to figure it out to keep me busy. At least I am exempt from homework for the time being... I will probably have it done tonight, it will just be a really tedious thing for me to do.

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LOL I think there isn't enough paper on Earth to write that number. It is approximately 4x102385, which is a lot of digits.

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LOL I think there isn't enough paper on Earth to write that number. It is approximately 4x102385, which is a lot of digits.

Actually, this is the number:

4038997629787155339700863409815084778394498166775976374862318662815021844263163724409589991283112221957087037127264409252982112748591787717033830403441930283161011881290431641966980623569028664868962702914864744551077531848115736772683548758847258321094808160079292956552763171104067984120533836065664635950242364928442451805995078317248461140444139995818842326862989533584638540917303432618956468436267462217689897536939221538008683721591946120333532143917872449136148108372559491267886787639350432567049929505139561975168349141248659914132248759237997505419159471214523173970710571263045668863231323715937900821485506870729657531757026555737371294825429353175800946829026948092511256737220542210787053051595802981233109856012113525552509973235479897937695548807826632854936270847693205577465760839058922819952696676524973128629373786196564822754641929042959146243903855562489356161956878595415082692189276329429991504770124701085279239460876288448740109138574892062762521143251789856063997453896592241444435083741307994418053089747011639244992143617911287606647084965258198883225653388806207929500332230594182854932910480899682575200047468631366224756184671205687777355791309481664752205737723827605017299803707184630307441302672768508598302249090453749312846375484742763396446462760789222817645292649569226868978755368552822174910148014846327742218968086229060583051969616187683845992803504299049605854491308472202616225188587696208053086463207413261782612698498484353406811946592391520876834837681364361483077335648507177704989176676017490814214154945785456307067444808828699697448178044358744486150076115286258469486513402087248384068655658114518474837867145754599634609879861608173455937726377253437847223098072299681760066838942906126088647741119141414552489886289568286295961338739388592134458987217604566798319860335993725331565539619297067041635560635329536488930786913392026253692233350241045325999643532468824953294370688166093949278863664041795436910656750167596038501554362222214884786870393545144578906190448059134680891645361639347000232719153886678836525568811533800230929254497238314075866436365607455976085809437067430000427425918303638570263277678578732590453700918386680277827005016588188884730521045514996708836288180634799955911110684992623893342705163686819347170019922026233208577169337941687350526454980398188591375023783468711359732633600493563136998276100001

It is exactly 2386 digits. This is a doable question. It will just take me a while

I will post a picture of the work once I finish it. Currently calculating 3^64 out by hand.

Now calculating 3^128

Edited by Lightmeow
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It is exactly 2386 digits. This is a doable question. It will just take me a while

I am seriously impressed!

Note by the way that the number of digits is $\log_{10}(3^{5000}) = 5000 \log_{10} (3) = 2385.6062...$ according to Wolfram Alpha, which checks out.

Edited by wtf
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Just finished 3^128...

Moving on to 3^256.

This is honestly the most retarded thing I've ever done.

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Impressive! Keep going!

Are you really good at this sort of thing in general? Do you have strong mental calculation skills and concentration?

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Impressive! Keep going!

Are you really good at this sort of thing in general? Do you have strong mental calculation skills and concentration?

Hmm... Seems little point to continue on, but I will, because why not.

Any one could do this if they tried, although I am pretty good at mental calculation and concentration.

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Hmm... Seems little point to continue on, but I will, because why not.

Any one could do this if they tried, although I am pretty good at mental calculation and concentration.

You'll certainly hardwire yourself for mental maths if you get to the end of this. Everything like that after will be a breeze. Eddy Izzard has just done 27 full marathons in 27 days I think you'll gain something from it, even though at first sight it seems pointless. Sustained concentration and focus are good traits to acquire.

Edited by StringJunky
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Hmm... Seems little point to continue on, but I will, because why not.

Any one could do this if they tried, although I am pretty good at mental calculation and concentration.

I notice you don't seem to write down your carries. Do you keep them in your head as you're computing the sum of the next column? Or do you write them down on a different piece of paper?

This is very impressive work. I am even more interested in the teacher-student dynamic here. They clearly recognized that you have a special aptitude for this kind of work. A few hundred years ago mathematicians and physicists had to do this all the time. I imagine Newton doing something similar when he worked out that the motion of the moon fit his theory of gravity.

This particular kind of skill, organization, and focus at hand calculation is very rare these days.

Edited by wtf
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Actually, this is the number:

4038997629787155339700863409815084778394498166775976374862318662815021844263163724409589991283112221957087037127264409252982112748591787717033830403441930283161011881290431641966980623569028664868962702914864744551077531848115736772683548758847258321094808160079292956552763171104067984120533836065664635950242364928442451805995078317248461140444139995818842326862989533584638540917303432618956468436267462217689897536939221538008683721591946120333532143917872449136148108372559491267886787639350432567049929505139561975168349141248659914132248759237997505419159471214523173970710571263045668863231323715937900821485506870729657531757026555737371294825429353175800946829026948092511256737220542210787053051595802981233109856012113525552509973235479897937695548807826632854936270847693205577465760839058922819952696676524973128629373786196564822754641929042959146243903855562489356161956878595415082692189276329429991504770124701085279239460876288448740109138574892062762521143251789856063997453896592241444435083741307994418053089747011639244992143617911287606647084965258198883225653388806207929500332230594182854932910480899682575200047468631366224756184671205687777355791309481664752205737723827605017299803707184630307441302672768508598302249090453749312846375484742763396446462760789222817645292649569226868978755368552822174910148014846327742218968086229060583051969616187683845992803504299049605854491308472202616225188587696208053086463207413261782612698498484353406811946592391520876834837681364361483077335648507177704989176676017490814214154945785456307067444808828699697448178044358744486150076115286258469486513402087248384068655658114518474837867145754599634609879861608173455937726377253437847223098072299681760066838942906126088647741119141414552489886289568286295961338739388592134458987217604566798319860335993725331565539619297067041635560635329536488930786913392026253692233350241045325999643532468824953294370688166093949278863664041795436910656750167596038501554362222214884786870393545144578906190448059134680891645361639347000232719153886678836525568811533800230929254497238314075866436365607455976085809437067430000427425918303638570263277678578732590453700918386680277827005016588188884730521045514996708836288180634799955911110684992623893342705163686819347170019922026233208577169337941687350526454980398188591375023783468711359732633600493563136998276100001

It is exactly 2386 digits. This is a doable question. It will just take me a while

I will post a picture of the work once I finish it. Currently calculating 3^64 out by hand.

Now calculating 3^128

I exaggerated, but you haven't done all the multiplications yet, it'll take a fair amount of paper. Maybe you can hand it in on a memory stick. You will have about sum 1...2385 digits or about 2844112. Actually, its more since you are calculating 5000 times.

Edited by EdEarl
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Title says it all. I've been challenged by my teacher to do this. I am currently doing it out by hand, it it is extremely tedious and annoying. Are there other more efficient ways to do this?

(P.S. I know I can put it into wolfram, I do have the answer. I just need to figure it out by hand)

Seems you could use the rules of exponents to shorten the work. ax*ay=ax+y so 32500*32500=35000 I'm sure you can shorten it more with a little thought, but I think that gets the idea across.

Started thinking about Ed's mention of the number of written numerals and paper used in light of my suggestion so I worked out 310 long as L. Meow is doing and then short as 35*35.

The long way I count 19 lines and 39 numerals and my short way I count 15 lines and 33 numerals. A modest improvement but an improvement none-the-less.

PS I didn't write down the carries digits so none are in my counts.

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A few hundred years ago mathematicians and physicists had to do this all the time. I imagine Newton doing something similar when he worked out that the motion of the moon fit his theory of gravity.

This particular kind of skill, organization, and focus at hand calculation is very rare these days.

Calculators were hired sometimes to perform calculations similar to these. I imagine Babbage was motivated to make his calculating engines by similar requirements.

PS

My wife, a teacher, suggested your teacher may have assigned this project because he/she needed to keep you busy while teaching less gifted students. If you want more challenging work, you might ask your teacher if you can do another project, e.g., perhaps research for science fair, with a report. If 35000 was assigned by your math teacher, emphasize the math you will use on the project, or if science teacher emphasize the science. Just a thought, if you find your current assignment too boring.

Edited by EdEarl
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Would it be cheating to write a computer programme to do it for you?

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Would it be cheating to write a computer programme to do it for you?

Yah, it would be. I have to do it all out by hand.

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Yah, it would be. I have to do it all out by hand.

If you want extra merit (and frankly I find it amazing you are doing it at all) - show the minimum number of calculations needed to do the calculation and the Order for this form of calc

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Correct me if I am wrong but

3^5000 = 9^2500

Edited by michel123456
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((((((((((((((((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

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=81^1250

Edited by michel123456
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81^625 not =81^624 + 81

33 = 32*3 not 32+3

Edited by EdEarl
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=6561^625

=6561^624 + 6561

Correct me if I am wrong but

3^5000 = 9^2500

step back

9^2500 = 10^2500 minus ?what?

81^625 not =81^624 + 81

33 = 32*3 not 32+3

Right. I corrected my post in the meanwhile.

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=81^1250

in fact (3^x)^(5000/x) but those representations don't really help

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step back

9^2500 = 10^2500 minus ?what?

At least it shows directly that the answer has less than 2501 decimals

The question resumes to

(x-1)^n

Maybe through a Pascal triangle ?

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9^2500 = 10^2500 minus ?what?

That is not a simple question - if you approach a simpler version algebraically it becomes clear

9^5 = 9*9*9*9*9

$10^5 = (9+1)^5 = (9+1)*(9+1)*(9+1)*(9+1)*(9+1)$

$= 1(9^5*1^0)+5(9^4*1^1)+10(9^3*1^2)+10(9^2*1^3)+5(9^1*1^4)+1(9^0*1^5)$

all the powers of one and figures to the power zero become 1

$= (9^5)+5(9^4)+10(9^3)+10(9^2)+5(9)+1$

the coefficients to the powers of 9 are obviously from pascals triangle and can be readily calculated - but it is still a lot of work. You would end up with a substraction from 10^2500 with 2499 terms being removed. And each of these terms being removed will require calculating and some will be considerably bigger than 9^2500. For instance the first negative term will be 2500*9^2499 - the second negative is much worse 31237500*9^2498.

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

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