Exponentiation, Its Derivative Operations, & Properties of Those Operations

Like multiplication is simply an operation that expresses a sequence of successive additions, exponentiation is an operation that expresses a sequence of successive multiplications.

Multiplication as addition:               5(5) or 5 x 5 = 5 + 5 + 5 + 5 + 5 = 25

Exponentiation as multiplication:    5^5 = 5 x 5 x 5 x 5 x 5 = 3125

Exponents – often known as ‘powers’ – are numbers that indicate how many times a base number may be multiplied by itself. This operation can be written in two standard ways:

          (read as “two to the third power”)


2^3 (read the same way)


This second method of writing exponentiation (i.e. that with a caret sign) is used in typing when superscripting (as in the first method) is unavailable or cumbersome.

For instance, the number 4^3 instructs you to multiply four by itself three times. The base is the number being raised by a power, whereas the exponent (or power) is the superscript number (or expression) above it (or the number/expression after the carrot sign).

It is worth noting that the power of two is also known as “squared” and the power of three is known as “cubed.” Sometimes, though rarely, the fourth power is called “hypercubed.” All other powers have no special designations.


The base and/or exponent in exponentiation can also be “expressions” or “operations” themselves and not only numbers. Take some of the below as examples of more complex exponentiation expressions.


(read as “two to the power of x”)

(read as “four to the power of three squared”)


(read as “x to the power of 2” or “x squared”)


(read as “three to the power of the quantity 7x minus 4”)


(read as “the quantity 5 minus 4/3s to the power of the quantity 3 minus one half”)


(read as “the quantity of x minus y to the power of a”)

exp 3.PNG
exp 4.PNG

Some Rules for Exponentiation

Rooting, The Inverse Operation of Exponentiation

The Laws of Exponents