# How do exponents raised to another exponent work?

Mar 29, 2016

See explanation...

#### Explanation:

One way of visualising basic positive integer exponents is as repeated multiplication:

${a}^{n} = {\overbrace{a \times a \times . . \times a}}^{\text{n times}}$

Then we find:

${a}^{m} \times {a}^{n} = {\overbrace{a \times a \times . . \times a}}^{\text{m times" xx overbrace(a xx a xx .. xx a)^"n times}}$

$= {\overbrace{a \times a \times . . \times a}}^{\text{m + n times}} = {a}^{m + n}$

$\textcolor{w h i t e}{}$
The next level of complexity is when a value that has been raised to an exponent is raised to another exponent:

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

For example, ${\left({2}^{2}\right)}^{3} = {4}^{3} = 64$

$\textcolor{w h i t e}{}$
The next level is where a value is raised to a value that has been raised to an exponent:

Note that ${a}^{{b}^{c}}$ is evaluated from right to left, not left to right.

So:

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

For example, ${2}^{{2}^{3}} = {2}^{8} = 256$