3.3 - Integer Exponents

In the following two sections, section 3.4 on multiplying expressions and section 3.5 on dividing expressions, it will be necessary to understand exponentials with integer exponents. For that reason we study integer exponents now. Later, when we study logarithms, we will need to understand exponents that are not integers. Click here to go to that topic.

When the exponential notation, bⁿ, is first introduced to students, it is defined to mean repeated multiplication:

where there are a total of n factors of b, and n can be any natural number, 1, 2, 3, …. The number b is called the base, n is called the exponent, and we say that we are “raising b to the n^th power” (except when n is 2 we say that we are “squaring b” and when n is 3 we say that we are “cubing b”). Here are some examples showing exponentials and what they mean:

2⁴ = 2 · 2 · 2 · 2 = 16
(−2)⁴ = (−2) · (−2) · (−2) · (−2) = 16
− 2⁴ = − 2 · 2 · 2 · 2 = −16

Note on the last example: this expression is considered to be the same as 0 − 2⁴ and, since exponentiation has precedence over subtraction, to be the same as − (2⁴).

Exponentials with the same base b have these three properties:

Multiplication property

Division property

Exponentiation property

The following three examples show why these properties are true for any positive integer exponents:

Let’s assume that these properties can be generalized to exponents that are not necessarily positive integers. Then we can also give meaning to zero and negative integer exponents. Here’s how:

The meaning of b⁰ : Let n = m in the division property. This gives:

On the other hand the numerator and denominator are equal:

Putting this together we find that b⁰ = 1. In other words any base raised to the 0^th power equals 1.

The meaning of b^{− n} : Let m = 0 in the division property. This gives:

On the other hand the numerator equals 1:

Putting this together we find that:

In other words any base raised to a negative power is the reciprocal of the same base raised to the corresponding positive power.

Notes:

An especially useful case is when the exponent is −1. The result is the reciprocal. Here are some examples:

Here in detail is how the −1 exponent produced the reciprocal in the last example:
We can’t raise zero to a negative power because then we get division by zero.

Exponentials whose bases are products or quotients

An exponential whose base is a product can be expanded like this:

Here is the proof:

Similarly, an exponential whose base is a quotient can be expanded like this:

Here is the proof:

Here is an example of an expansion that combines several of these concepts:

Algebra Coach Exercises

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Multiplication property
Division property
Exponentiation property