How do you multiply double exponents?

[parallaxicality]

I've been having an issue with Wikipedia; I need to somehow explain to Joe Public in layman's terms what the number 10^10^10^122 means. Well, I can say that it is 1 followed by 10^10^122 zeroes. I know that 10^10^100 is a googolplex, but how many times a googolplex is 10^10^122? I initially thought that it was 10^22 times a googolplex, but I think it might be 10^10^22 times a googolplex. Er, help?

[antoniseb]

You could test it with smaller numbers ...

2^2^3 = 256.
2^2^5 = 4,294,967,296.

The ratio is: 2^24 ~= 2^2^4.7

So I think the answer is that 10^10^122 is almost 10^10^122 times bigger than a Googolplex.

[swampyankee]

10^10^10^122 may be easier to sort out with some parentheses:

((10^10)^10)^122

Expanding the innermost parentheses:

(10 000 000 000^10)^122

Repeat as needed, but I think you'll end up with 1 followed by 10^244 zeroes.

In math, parentheses are your friends

eta

Parentheses are, indeed, your friends. If the correct way to parse something like this is from the top down:

10^10^10^122 should be parsed as

10^10^(1 followed by 122 zeroes)

10^(1 followed by 10^122 zeroes)

I've probably messed this one up, too.

I will now, officially, give up until I dig up a text book that describes the rules instead of relying on a memory that's increasing like a steel sieve.

[tony873004]

2^2^3 = 256.

10^10^10^122 may be easier to sort out with some parentheses:
((10^10)^10)^122

By this logic, antoniseb should have got 64 instead of 256.
256 requires
2^(2^3)

My TI-84 says that 2^2^3 = 64, as does writing a short computer program in Visual Basic:

x=2^2^3
print x

gives 64. But Google calculator says 2^2^3 = 256

Only parenthesis can get you beyond this ambiguity

[a1call]

If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:
...

Special cases

[WayneFrancis]

By this logic, antoniseb should have got 64 instead of 256.
256 requires
2^(2^3)

My TI-84 says that 2^2^3 = 64, as does writing a short computer program in Visual Basic:

x=2^2^3
print x

gives 64. But Google calculator says 2^2^3 = 256

Only parenthesis can get you beyond this ambiguity

I think that is because of the way you enter the equation into your TI-84

typing 2 ^2 ^3 your doing left to right which is not the correct order of precedence.

2²³ is 256

You'd have to do 2^(2^3) for your calculator.

[grapes]

I know that 10^10^100 is a googolplex, but how many times a googolplex is 10^10^122? I initially thought that it was 10^22 times a googolplex, but I think it might be 10^10^22 times a googolplex. Er, help?

So I think the answer is that 10^10^122 is almost 10^10^122 times bigger than a Googolplex.

In a way, I think that's right.

But that tells us nothing at all about how big 10^10^122 is.

Another way of putting it is, 10^10^122 is googolplex^(10^22):

googolplex^(10^22) =
(10^10^100)^(10^22) =
10^(10^100 * 10^22) =
10^(10^122)

So, 10^10^122 equals a googolplex times itself 10,000,000,000,000,000,000,000 times

[Bogie]

When you raise a power to a power, don't you multiply the powers? If so, then ((10^10)^10)^122 gives 10^12200?

[grapes]

When you raise a power to a power, don't you multiply the powers? If so, then ((10^10)^10)^122 gives 10^12200?

Yes and no. ((A^x)^y)^z equals A^(xyz) but when you see A^x^y^z it typically means A^(x^(y^z)), which might be a much larger number. As the OP says, a googol is 10^100, and a googolplex is 10^googol, so a googolplex is 10^(10^100) which is 1 followed by 10,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000 zeroes, whereas (10^10)^100 has only 1000 zeroes.

[utesfan100]

I thought exponents were associative.

(2^2)^2=4^2=16
2^(2^2)=2^4=16

[Hetman]

(10^10)^10 = 10000000000^10 = 10^100
10^(10^10) = 10^10000000000

[swampyankee]

By this logic, antoniseb should have got 64 instead of 256.
256 requires
2^(2^3)

My TI-84 says that 2^2^3 = 64, as does writing a short computer program in Visual Basic:

x=2^2^3
print x

gives 64. But Google calculator says 2^2^3 = 256

Only parenthesis can get you beyond this ambiguity

Most programming languages parse arithmetic statements with the same operator -- like 1*2*3*4*5.... from left to right. They parse exponents from right to left, so 2^3^4 is parsed as 2^(3^4), or 2^81. The way I parenthesized it would be (2^3)^4, which is 8^4, which is not the same. I think the right-to-left method is right, and I erred above.

[WayneFrancis]

the right to left rule is correct for order of precedence and another way to think about it is like this

2^3^4^5

is since exponents are done first 2^3 can't be done because 3 is raised to the power of 4 and well 4 has a power but 5 doesn't....
actually Right to Left rule is more simple to remember.

The exception being when there is parentheses involved. IE (2^3)^4 is not the same as 2^3^4
The former is only 4096 where the later is 2,417,851,639,229,258,349,412,352
2^3^4 and 2^(3^4) are the same. I hate useless parentheses

[Bogie]

OK, I have to admit to a mental block here .

To make it simple, we are raising x to three levels of power as follows"
x^2^3^4

For x=2:
x^2^3^4=x^24=((X^2)^3)^4=(x^6)^4=(x2)^12=(x^8)^3=
16,777,216

My mental block, if it is a mental block is about raising a power to a power to a power which I think is accomplished by multiplying the powers. *It doesn't matter which order you do the multiplying, does it?

x^24=2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2 *2=16,777,216

((x^2)^3)^4=*
2*2=4 *
4*4*4=64 *
64*64*64*64=16,777,216

(x^6)^4=
2*2*2*2*2*2=64
64*64*64*64=16,777,216

(x^2)^12=
2*2=4
4*4*4*4*4*4*4*4*4*4*4*4=16,777,216

(x^8)^3=
2*2*2*2*2*2*2*2=256
256*256*256=16,777,216

[grapes]

X^2^3^4 is not x^24, it's X^(2^(3^4)), which is X^(2^81), which is X^2417851639229258349412352

[Bogie]

Can you show me the steps using x=2? And maybe a link to the rule?

[grapes]

That would be 2^2417851639229258349412352. The rule was linked by a1call in post #5, labeled "special cases"

[Bogie]

So x^2^3^4 is not ((x^2)^3)^4, it is x^(2^(3^4) then?
And if I could only figure it out from there I would be happy.

So let's see ... how do you get X^(2^81) from that. I other words, can you show me the steps instead of saying that would be 2^2417851639229258349412352?

[grapes]

3^4 is 81

[Bogie]

Oh, so it is, lol. I was still trying to do 3*4. From there is see ... 2^81 must be 241785163922925834941235, and so then when x=2, it comes out 2^2417851639229258349412352. Slightly higher than what I get by failing to follow the "special cases" rule. Thanks.

[grapes]

You're welcome!

Now, where's parallaxacality?

[parallaxicality]

Over at

http://en.wikipedia.org/wiki/Orders_...e_%28length%29

Hope I managed to make sense of all this...

[grapes]

10^10^10^122 metres = 10^10^10^122 yottameters? That's gonna get you in the tabloids!

[Bogie]

Now there is a rule that will help, lol. The "unitless" rule should be a big boost when sorting out units of measure. Just multiply your answer by 10^10^10^122 and use any units of measure you like. Where was that when I was in school?