# Why exponent of exponent multiplied?

If I have an expression like ${2^2}^3$, then why are the powers multiplied? In other words, why is the above expression equal to $2^6$ not $2^8$?

• The expression you wrote doesn't compile in LaTeX. Do you mean 2^2^3 or $2^{2^3}$ or $(2^2)^3$? Commented Apr 4, 2015 at 9:26

Exponentiation is not associative, there is a real difference between $(2^2)^3$ and $2^{(2^3)}$.
As @Rolf Hoyer writes, exponentiation is not associative, hence ${2^2}^3$ would result in an ambiguity: for this reason, when we write $2^{2^3}$ it is always meant to be $$2^{(2^3)}=2^8.$$ Otherwise, if we mean $$(2^2)^3=2^{2\cdot 3}=2^6$$ brackets must appear.
In general, $$x^{a^{b^{c^{\cdots}}}} = x^{{\left(a^{\left(b^{\left({c^{(\cdots)}}\right)}\right)}\right)}}\ne (((x^a)^b)^c)^{\cdots}=x^{a\cdot b\cdot c\cdot \ldots}$$ while, for example, $${{\left(x^{(a^b)}\right)}^c}^{\ldots} \qquad\text{or}\qquad x^{{\left({(a^b)}^c\right)}^{\ldots}}$$ must be explicitly denoted with brakets.