# Power tower of $10$

Is $10^{10^{10^{10}}}$ the same thing as $10000000000^{10000000000}$? Thought this from writing it as $(10^{10})^{(10^{10})}$. Can it be done like this or should it be written another way?

• Exponentiation is not associative in the sense that, in general, $(a^b)^c \ne a^{(b^c)}$. – Xander Henderson Mar 30 '18 at 15:42

We have $$x =10^{10^{10^{10}}}=10^{10^{10000000000}}$$

Now, the order in which the operations are carried out is from the top, so no, the following operation would be to compute the exponent, which is $a=10^{10000000000}$ (a 1 followed by 10000000000 zeroes) and finally to compute $$x=10^a.$$

What you did is $(10^{10})^{(10^{10})}$ which actually equals $10^{10^{11}}$.

• ah thanks i see – Henry Lee Mar 30 '18 at 15:45
• Great. As the comment from @Xander Henderson states, problem here is associativity. Be sure to accept an answer to avoid going into the unanswered section. – Tal-Botvinnik Mar 30 '18 at 15:47

No, exponentiation is not associative. (a^a)^a is equal to a^(a * a), which is not equal to a^(a^a). Since (a^a)^a has the alternative form a^a*a, there is no need to write it (a^a)^a. Thus, the ambiguity of a^a^a being interpreted as (a^a)^a or a^(a^a) is resolved in favor of a^(a^a), since (a^a)^a already has another way of being written and does not need this form. Thus, when interpreting a tower of exponentiation, the expression is parsed right to left. So to evaluate 10^10^10^10, you first evaluate the last part: 10^10^10^10. Once that part is calculated, it's used to evaluate the next part to left: 10^10^(10^10). So 10^10^10^10 = 10^(10^(10^10)).

• You've a good username for this problem. And you acccumulate c's. – Ethan Bolker Mar 30 '18 at 15:55
• @Acccumulation: The phrase "So to evaluate 10^10^10, you first evaluate..." is missing another 10, I think it should be 10^10^10^10. – Mathew Mahindaratne Mar 30 '18 at 16:29

No. Exponents are always evaluated upside down. So you cannot just put bracket here and there. For instance

$${{{{2}^2}^2}^2}={{{2}^2}^{(2^2)}}=2^{(2^4)}=2^{16}$$

$${{{{2}^2}^2}^2}\ne {{(2^2)^2}^2}$$

Note that

$$\Large{10^{10^{10^{10}}}=10^{10^{\overbrace{10000000000}^{\text{10 zeros}}} }=10^{\overbrace{10000...00000}^{10^{10} \text{zeros}}}}$$

This breaks down quickly: $${(2^2)}^2=16=2^{(2^2)}$$ But $${(3^3)}^3={27}^3=19,683$$ $$3^{(3^3)}=3^{27}=7,625,597,484,987$$