What is the reasoning behind this exponents question?

Question

What is $3^{3^{3}}?$

Plugging $3^{3^{3}} $into the calculator gives 7625597484987. I believe because this implies that $3^{3^{3}}=3^{27}$, is this true?

And plugging $(3^{3})^{3}$ gives 19683, because $ (3^{3})^{3}=3^{3}\times 3^{3}\times 3^{3}=3^{9}=19683$

So which one is the correct answer, and why?

Answer 1

Unlike addition and multiplication, exponentiation is not associative:

• $(a+b)+c=a+(b+c)$
• $(a\times b)\times c=a\times (b\times c)$

but

• ($a$^$b$)^$c\ne a\!$^($b$^$c$), more commonly written as: $\left(a^b\right)^c \ne a^{\left( b^c \right)}$

This means there's no risk in simply writing "$a+b+c$" or "$a \times b \times c$" since the order in which you perform the operations doesn't matter in both cases.

For exponentiation this is not the case and writing "$a$^$b$^$c$" is ambiguous, but we do have:

$$\color{blue}{\left(a^b\right)^c = a^{bc}} \ne a^{\left( b^c \right)}$$

Because we have this property (in blue), it's common to interpret $a^{b^c}$ as $a^{\left( b^c \right)}$ but if you want to avoid confusion, you can always add the parentheses.

Answer 2

The accepted answer is not completely explicit, but the reason why $a^{b^c}$ means $a^{(b^c)}$ is not purely mathematical, but conventional. We chose to use the exponentiation notation in this manner, namely the whole superscripted expression is the exponent, like this: $a^\boxed{b^c}$. And it is important to note that we can make such a convention because it is unambiguous; you can always make it clear by the superscript formatting what is supposed to be the exponent.

As StackTD said, addition and multiplication are associative, and hence it is not problematic to use a linear notation and omit brackets. In contrast, exponentiation is not associative, so it makes sense to use this kind of non-linear notation so that we can still omit brackets! You can imagine the superscript formatting as literally standing in for the bracketing.

Similarly for other notation issues; the first step is to know precisely what is the convention for interpreting the notation. For example, $-2^3$ is by convention interpreted as $-(2^3)$ because exponentiation is given higher precedence than negation, and $2+3·4 = 14$ because $·$ has higher precedence than $+$.

Answer 3

a^b^c and (a^b)^c are different because (a^b)^c is the same as a^b multiplied by itself $c$ times. a^b^c is rather confusing, so you should use parentheses like this: a^(b^c), which is $a$ multiplied by itself b^c times. a^b^c should have parentheses just to avoid confusion.