# Why is $2^{2^{2^n}}$ not equal to $16^n$?

Why is $a^{b^{c^d}}$ not equal to ${(a^{b^c})}^d$ (for positive n)?

For example, WolframAlpha seems to say that $2^{2^{2^n}}$ is not equal to $16^n$.

• Possible duplicate of What is the solution of $2^{2^{2^2}}$? – Ross Millikan Nov 23 '16 at 19:55
• @Did How is 2^2^2^n ambiguous? It's clearly 2^(2^(2^n)) – Max Nov 23 '16 at 19:58
• @Max And what happens if somebody insists that it is "clearly" ((2^2)^2)^n? You call the Pope for their advice? – Did Nov 23 '16 at 19:59
• @Max Not every author uses this convention (which seems to be originating from a programming context rather than from mathematics). Hence: ambiguous. Hence: to be avoided. – Did Nov 23 '16 at 20:02
• @Max, how is this priority rules? Exponentiation has higher priority then exponentiation? This is not a case of $2+2^2$ for priority to decide, this is a question of left/right associativity. I do agree that right associativity is usual here (I've never seen someone treating it as left associative) but this is not obvious, it is a convention (differing from convention for subtraction, for example). – Ennar Nov 23 '16 at 20:08

It is purely a matter of notational convention that $a^{b^c}=a^{(b^c)}$ rather than $(a^b)^c$, but it's a convention that makes sense: There would be no point in using the notation for the second convention, since it would be easier to simply write $a^{bc}$ -- and likewise for $a^{b^{c^d}}$ if the notation meant $a^{b^{c^d}}=(a^{b^c})^d=((a^b)^c)^d=a^{bcd}$. One reason for adopting it as a convention is that the unambiguous notation

$$a^{\left(b^{\left(c^d\right)}\right)}$$

takes up way more room.

The problem is here what you mean by

a^b^c


For example: $$(2^3)^2 = 8^2 = 64 \\ 2^{(3^2)} = 2^9 = 512$$ Or $$(2^{(2^2)})^3 = 16^3 = 4096\\ (2^2)^{(2^3)} = 4^8 = 65536$$ So there is an example with $n=3$.

You want to use paratheses to make it clear what you want to do.

In fact recall the rule that $$(a^b)^c = a^{b\cdot c}$$

The operation ^ is not associative, i.e. in general $a^{(b^c)}\neq (a^b)^c$. The usual convention is $a^{b^c}=a^{(b^c)}$.

Although I think Barry Cipra's answer should be the canonical one, I think there is also an argument against the OP's proposal on typographical grounds: if the intent of $a^{b^c}$ was to convey $(a^b)^c$, then there would be no justification for making the $c$ smaller, since it is then a top-level exponent and not a superscript within a superscript: it should look like ${a^b}^c$ rather than $a^{b^c}$. The fact that the $c$ is intentionally typeset smaller than $b$ establishes the intent that the $c$ is an exponent within a term that is already in smaller type size, namely the exponent of $a$.

Of course, semantics need not always follow syntactic structure literally. For instance, there is a practical limit to how small one can make subscripts, so in a tall tower of exponents we might see exponents at different heights with the same type size. But I can see no reason for exponents at the same height to have different sizes.

The mistake people make usually is they go from down to up rather than coming from up to down. $${16^n}$$ is nothing but $${(2^n)(2^n)(2^n)(2^n)}$$.

On the other hand, While solving for $$2^{2^{2^n}}$$ You will have to come down by giving the upper powers to the 2's which are lying below.

For example $$2^{2^{2^2}}=(2^{16})$$

If we use your terminology it will be 256 which is false.

• What does "terminology" mean in the last sentence? – Erick Wong Nov 23 '16 at 21:08
• @ErickWong, I just wanted to say that if the OP will use the method he is asking, the answer will be wrong. – Vidyanshu Mishra Nov 23 '16 at 21:11
• Perhaps "notation" makes more sense in the last line. – Simply Beautiful Art Dec 29 '16 at 20:42

Note: $a^{b^{c^d}} = a^{b^{(c^d)}} \neq {(a^{b^c})}^d$