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Why does $\;\large3^{3^{3^3}}\;$ evaluate to a larger number than $\;\large 27^{27}$?

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The first one can be written $3^{3^{3^3}} = 3^{3^{27}} = 3^{7625597484987}$. The other one is $27^{27} = (3^3)^{27} = 3^{3\cdot 27} = 3^{81}$. Clearly the first one is much bigger. –  Jeppe Stig Nielsen Apr 1 '13 at 8:59
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3 Answers

up vote 14 down vote accepted

One is $$(3^3)^{3^3}$$ while the other is $$3^{3^{3^3}}$$

What is the difference between $$3^{3^3}$$ and $${(3^3)}^{3}\text{ ?}$$

In the first one, you make $3^3$, and then raise $3$ to that power.

In the other, first take $3^3$, and raise that to the power of $3$. See the difference?

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Big letters only used for visibility:

$$3^{3^{3^3}} = 3^{3^{27}} \ne 3^{3^4} = 3^{3 \cdot 3^3} = (3^3)^{3^3} = 27^{27}$$

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In short: $$\Large 3^{3^{3^3}} = \color{blue}{\bf 3^{3^{27}}} \neq 27^{27} = 3^{{3\times 3}^3} = \color{blue}{\bf {3^3}^4}$$

ADDED: The expression to the left is called a tetration. You can see it tetration compared to exponentiation and other operations, plus lots of information about such expressions at the linked Wikipedia entry.

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User 70366 - Do you understand that in general, $(a^b)^c = a^{bc} \neq a^{b^c}$, and so, for example in this case, we need only look at $(3^3)^3 = 3^{3\times 3} \neq 3^{3^3}$? The same applies to tetrations of more than three exponents. –  amWhy Apr 1 '13 at 2:31
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The most paradigm-breaking thing in this answer is that it has shown me that I can change colors in MSE. Although obvious, I never thought about it. –  Marra Apr 1 '13 at 2:33
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Thanks, @Gustavo: Color is really useful at times! Is the use of color worth an upvote? ;-) –  amWhy Apr 1 '13 at 2:35
    
Right away, sir! –  Marra Apr 1 '13 at 2:37
    
@GustavoMarra You wanna change that to "ma'am"! –  Pedro Tamaroff Apr 1 '13 at 3:30
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