What's the difference between $3^{3^{3^3}}$ and $27^{27}\;$?

Question

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

Answer 1

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?

Answer 2

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.

Answer 3

Big letters only used for visibility:

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