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# Sequence when exposing

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X = 10^10^100
What does that mean?
$$X=(10^{10})^{100}$$                 $$X=10^{(10^{100})}$$

asinus  Sep 5, 2017

#3
+2075
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Hmmm... I'm sorry for not explaining the exponent towers earlier. I'll try my best to give a logical explanation on why exponent towers should be evaluated from the top downward despite it possibly being counterintuitive. To answer your original question, $$10^{10^{100}}=10^{\left(10^{100}\right)}$$

Let's use the variables a, b, and c and suppose that $$a^{b^{c}}=\left(a^{b}\right)^c=a^{bc}$$, then the existence of exponent towers would be completely unnecessary as only one level of the exponent would ever be necessary; there must be a reason for such notation to exist.

Another reason is that it is consistent with the order of operations. Exponents is a level below parentheses when it comes to priority, but the order of operations applys to the power, too. Here's an example:

$$2^{2+3}=2^5=32$$

$$3^{2*2}=3^4=81$$

And therefore, you must evaluate the exponent within the exponent. Does this make sense?

I learned today that even modern-day programs calculate have ambiguity in the matter. Excel 2013 is an example, as $$2^{3^2}$$ outputs 64 as opposed to 256. In the future,

TheXSquaredFactor  Sep 6, 2017
edited by TheXSquaredFactor  Sep 6, 2017
#1
+1

X = (10^10)^100 = (10,000,000,000)^100 = 10^(10*100) = 10^1,000.

X = 10^(10^100) = 10^(1 + 100 zeroes) = 10 ^ 10 ^ 100 = 10^googol.

Guest Sep 5, 2017
#2
+7095
+2

I think that when parenthesees aren't included, it makes the most sense to say....

a ^ b ^ c  =  a ^ ( b^c )

c  does not apply to  a  and  b  ,     c  applies only to  b  .

hectictar  Sep 6, 2017
edited by hectictar  Sep 6, 2017
edited by hectictar  Sep 6, 2017
#3
+2075
+1

Hmmm... I'm sorry for not explaining the exponent towers earlier. I'll try my best to give a logical explanation on why exponent towers should be evaluated from the top downward despite it possibly being counterintuitive. To answer your original question, $$10^{10^{100}}=10^{\left(10^{100}\right)}$$

Let's use the variables a, b, and c and suppose that $$a^{b^{c}}=\left(a^{b}\right)^c=a^{bc}$$, then the existence of exponent towers would be completely unnecessary as only one level of the exponent would ever be necessary; there must be a reason for such notation to exist.

Another reason is that it is consistent with the order of operations. Exponents is a level below parentheses when it comes to priority, but the order of operations applys to the power, too. Here's an example:

$$2^{2+3}=2^5=32$$

$$3^{2*2}=3^4=81$$

And therefore, you must evaluate the exponent within the exponent. Does this make sense?

I learned today that even modern-day programs calculate have ambiguity in the matter. Excel 2013 is an example, as $$2^{3^2}$$ outputs 64 as opposed to 256. In the future,

TheXSquaredFactor  Sep 6, 2017
edited by TheXSquaredFactor  Sep 6, 2017