Question

How do exponents raised to another exponent work?

Answer 1

See explanation...

Explanation:

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

`a^n = overbrace(a xx a xx .. xx a)^"n times"`

Then we find:

`a^m xx a^n = overbrace(a xx a xx .. xx a)^"m times" xx overbrace(a xx a xx .. xx a)^"n times"`

`=overbrace(a xx a xx .. xx a)^"m + n times" = a^(m+n)`

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The next level of complexity is when a value that has been raised to an exponent is raised to another exponent:

`(a^b)^c = a^(bc)`

For example, `(2^2)^3 = 4^3 = 64`

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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^((b^c))`

For example, `2^(2^3) = 2^8 = 256`