Order of powers

Question

Which of the following is $a^{b^c}$ equal to?

$a^{(b^c)}$ or
${(a^b)}^c = a^{bc}$

This is probably a really basic question but I can't seem to find the answer anywhere.

Answer 1

In a sense this is a problem of convention and notation. In mathematical literature the usual interpretation is the first one you mentioned. However if you write a^b^c in a calculator/computer the answer might vary depending on the model/software as the "precedence rule" for powers is not universally agreed upon.

Answer 2

It will be $a^{(b^c)}$ because ${(a^b)}^c = a^{bc}$ will yield a different result. For example:

let $a = 2$, $b = 3$ and $c = 4$.

$$2^{(3^4)} = 2^{81} = 2417851639229258349412352$$

while

$${(2^3)}^4 = 8^{4} = 2^{12} = 4096$$

Answer 3

It is $a^{(b^c)}$. The usual explanation is that $(a^b)^c$ can be easily written $a^{bc}$