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:
$a^b < a^1 \\ b$
But exponent function a^b has to be increasing so a>1 
So first solution:  a>1 and 0<b<1.  Examples:
$5^{\frac{1}{2}} < 5 \\ 120^{\frac{1}{12}} < 120$

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:
$\left(\dfrac{1}{2}\right)^{12} < \dfrac{1}{2} \\ \\ \left(\dfrac{12}{39}\right)^{16\frac{1}{2}} < \dfrac{12}{39}$