Question

write a power represented with a positive base and a positive exponent whose value is less than the base

Answer 1

You mean situation, when a^b < a.
There are two ways:

`a^b < a^1 \\ b<1`
But exponent function a^b has to be increasing so a>1
So first solution: a>1 and 0<b<1. Examples:

`5^{(1)/(2)} < 5 \\ 120^{(1)/(12)} < 120`

and second way:

`a^b < a^1 \\ b>1`
Exponent function a^b has to be decreasing, so 0<a<1
Second solution: a<1 and b>1. Examples:

`\left((1)/(2)\right)^(12) < (1)/(2) \\ \\ \left((12)/(39)\right)^{16(1)/(2)} < (12)/(39)`