Exponentiation is a binary operator that operates on a base and exponent value. The exponent operator is represented by writing the exponent value in superscript style. For discrete numbers, a base value to the power is the same as multiplying the base by itself times. For example, raised to the value is equal to multiplied by multiplied by as shown in the equation below.
The exponent operator models exponential growth and is related to the logarithm and radical operators. The relationship between the three operators and the nature of the exponent operator itself can be visualized by drawing a tree structure.
Shown below are examples that demonstrate the result of the exponent operator for discrete numbers.
Expression  Pattern 

Expression  Pattern 

Expression  Pattern 

The properties of the exponent operator are summarized in the table below. These properties can be used to manipulate math expressions into new forms.
Name  Property 

Zero Property  
Identity Property  
Negative Property  
Product Property  
Quotient Property  
Power Property  
Radical Property 
The exponential operator is closely related to the logarithm and radical operators. For example, given the scenario of exponential population growth, the relationship between the three operators is expressed below. Note, in this hypothetical scenario the initial population is represented with the value of .
Equation  Operator 

Exponent Given the growth rate and the time elapsed, the exponent operator returns the population. 

Logarithm Given the growth rate and the population the logarithm operator returns the time elapsed. 

Radical Given the time elapsed and the population the radical operator returns the growth rate. 
Taking the logarithm of an number is the inverse operation of exponentiation. The subscript number is the base of the logarithm and the expression is the operand.
The radical operator returns the nth root of the provided expression. The radical operator is an alternative way of writing a fractional exponent.
This article explores a technique of visualizing the three math operators: expononents, logarithms and radicals using the structure of a tree.