3.8 Exponents

An illustration shows the solar system. The sun, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune are labeled. — **Figure 3.44** Astronomical distances are written using exponents.Astronomical distances are written using exponents. (credit: “Our Solar System (Artist's Concept)” by NASA/Jet Propulsion Laboratory-Caltech/Public Domain)

Learning Objectives

After completing this section, you should be able to:

Apply the rules of exponents to simplifying expressions.

Sometimes, we look for shorthand when writing or expressing something that simply takes too long. The use of LOL and tl;dr. This shorthand only works if everyone reading the shorthand knows what it stands for. Using exponents is a similar instance. Writing out a long string of a number times itself over and over takes too much time, and eventually one would forget how many of the value has been written or read. For example, $8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8$ . There has to be a shorter and more efficient way to write 8 times itself 1, 2, 3….hmmmm, 19 times.

And that’s the role that exponents play in mathematics. They are shorthand for multiplying a number by itself a number of times. Without it, calculations would become a mess and we’d have to write a lot more.

Applying the Rules of Exponents to Simplify Expressions

Squaring a number is multiplying it by itself, and has that name because it is the area of a square with that side length. Cubing a number is finding the volume of a cube with that length of sides. That’s why we refer to $5 squared$ as five squared, or $10 cubed$ as ten cubed. Exponents represent that multiplication.

Let’s remind ourselves of the terminology associated with exponents and what exponents represent. Suppose you want to multiply a number, let’s label that number $a$ , by itself some number of times. Let’s label the number of times $b$ . We denote that as $a^{b}$ . We say $a$ raised to the $b$ th power. When we write or see $7 to the power 5$ , we call the 7 the base and we call 5 the exponent. What it represents is 7 multiplied by itself 5 times. This means exponents are used as a shorthand for repeated multiplications, where we write $7^{5} = 7 \times 7 \times 7 \times 7 \times 7$ . We would write $7 to the power 5$ and say seven to the fifth power.

The definitions of base and exponent make it possible to understand the exponent rules.

Product Rule for Exponents

The first rule we examine is the product rule, $a^{n} a^{m} = a^{n + m}$ . This rule means that when we multiply a base raised to a power times the same base to another power, the result is the base raised to the sum of the powers. To demonstrate, consider $9^{3} \times 9^{5}$ . If we apply the product rule to that we get $9^{3} \times 9^{5} = 9^{3 + 5} = 9^{8}$ . This can be tested by looking at the multiplications that are represented. The $9 cubed$ is 9 times itself 3 times, while $9 to the power 5$ is 9 times itself 5 times. Substituting those into $9^{3} \times 9^{5}$ we see $9^{3} \times 9^{5} = (9 \times 9 \times 9) \times (9 \times 9 \times 9 \times 9 \times 9) = 9^{8}$ , which is what the formula told us would happen.

These rules can be applied to unknowns too.

Quotient Rule for Exponents

The next rule we examine is the quotient, or division, rule.

This rule means that when we divide a base raised to a power by the same base to another power, the result is the base raised to the difference of the powers. To demonstrate, consider $\frac{14^{13}}{14^{6}}$ . If we apply the quotient rule to that, we get $\frac{14^{13}}{14^{6}} = 14^{13 - 6} = 14^{7}$ . This can be tested by looking at the division that is represented. Remember, $14 to the power 13$ is 14 multiplied to itself 13 times, while $14 to the power 6$ is 14 multiplied to itself 6 times. Substituting those into $\frac{14^{13}}{14^{6}}$ gives the following:

$\frac{4^{13}}{4^{6}} = \frac{4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4}{4 \times 4 \times 4 \times 4 \times 4 \times 4}$

We see here that there are a LOT of fours to be divided out.

$= \frac{4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4}{4 \times 4 \times 4 \times 4 \times 4 \times 4} = \frac{4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4}{1} = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$

What remains is 4 to the 7th power, $4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4^{7}$ .

All of the work above confirmed what the formula told us would be the result.

A natural consequence of the quotient rule is what it means to raise a non-zero number to the zeroth power. Let’s look at the simplification when the exponents are equal.

$\frac{3^{6}}{3^{6}} = 3^{(6 - 6)} = 3^{0}$

We know that a number divided by itself is 1, so $\frac{3^{6}}{3^{6}} = 1$ . From that is must be that $\frac{3^{6}}{3^{6}} = 3^{0} = 1$ . This provides the rule for a number raised to the power 0: $a not equal to 0$ .

Distributive Rule for Exponents

The next rule we look to is a distributive rule for exponents.

This means that when we have two numbers multiplied together, and that is raised to a power, it is the same as raising each of the numbers to the same power first, then multiplying. For example, ${(3 \times 7)}^{4} = 3^{4} \times 7^{4}$ . This can be explained using the definition of exponents and multiplying all the factors.

${(3 \times 7)}^{4} = (3 \times 7) \times (3 \times 7) \times (3 \times 7) \times (3 \times 7)$

We may change the order in which numbers are multiplied. This is the commutative property of the real numbers. This can be written as $3 \times 3 \times 3 \times 3 \times 7 \times 7 \times 7 \times 7$ . Using exponents, that shortens to $3^{4} \times 7^{4}$ .

This also works in the other direction, $a^{n} \times b^{n} = {(a \times b)}^{n}$ . Read this way, if we have one base raised to an exponent, and another base raised to the same exponent, we can multiply the bases and raise that product to the shared exponent. For instance, $7^{8} \times 11^{8} = {(7 \times 11)}^{8} = 77^{8}$ .

This distribution also works for quotients. A fraction raised to an exponent equals the numerator raised to the exponent divided by the denominator raised to the exponent. For example, ${(\frac{3}{5})}^{7} = \frac{3^{7}}{5^{7}}$ . Demonstrating this is similar to the previous rule.

Power Rule

In the previous two sets of rules, we’ve seen exponents applied to products and quotients. Now we look to exponents applied to other exponents. For example, ${(3^{6})}^{4} = 3^{(6 \times 4)} = 3^{24}$ . This can be explained by examining what the outer exponent does. We raise $3 to the power 6$ to the fourth power, so we multiply $3 to the power 6$ by itself 4 times, ${(3^{6})}^{4} = 3^{6} \times 3^{6} \times 3^{6} \times 3^{6}$ . Now if we apply the product rule for exponents, this becomes $3^{(6 + 6 + 6 + 6)} = 3^{24}$ .

Negative Exponent Rule

Up until now, we’ve only looked at positive exponents. The last exponent rule we look at is what negative exponents represent. Recall the quotient rule: $\frac{a^{n}}{a^{m}} = a^{(n + m)}$ . What would happen if the exponent in the denominator was larger than that in the numerator? For example, $\frac{4^{5}}{4^{7}}$ . If we apply the quotient rule, we obtain $\frac{4^{5}}{4^{7}} = 4^{5 - 7} = 4^{- 2}$ . We need to make sense of that negative exponent. To do so, we can expand the quotient and see what happens: $\frac{4^{5}}{4^{7}} = \frac{4 \times 4 \times 4 \times 4 \times 4}{4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4}$ . When we divide out common factors, only two factors of 4 are left in the denominator, as we see here: $\frac{1}{4 \times 4}$ . Using exponent notation, this is $\frac{1}{4^{2}}$ . Since $4^{-}^{2}$ and $\frac{1}{4^{2}}$ represent the same number, $\frac{4^{5}}{4^{7}}$ , they are equal. This demonstrates how negative exponents are defined.

The table below shows a summary of the exponent rules from this section.

Rule	Example	In Words
Product Rule $a^{n} a^{m} = a^{n + m}$	$8^{2} \times 8^{5} = 8^{7}$	A base raised to a power, times the same based raised to another power, is the base raised to the sum of the powers.
Quotient Rule $\frac{a^{n}}{a^{m}} = a^{(n - m)}$	$\frac{11^{14}}{11^{12}} = 11^{12}$	A base raised to a power, divided by the same based raised to another power, is the base raised to the difference of the powers.
Zero Power Rule $a^{0} = 1$ provided that $a not equal to 1$	$412^{0} = 1$	Any non-zero number raised to the zeroth power equals 1.
Distributive Rule, Multiplication ${(a \times b)}^{n} = a^{n} \times b^{n}$	${(14 \times 31)}^{9} = 14^{9} \times 31^{9}$	Exponents distribute across multiplication.
Distributive Rule, Division ${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$	${(\frac{62}{91})}^{8} = \frac{62^{8}}{91^{8}}$	Exponents distribute across division.
Power Rule ${(a^{n})}^{m} = a^{(n \times m)}$	${(5^{9})}^{15} = 5^{135}$	A base raised to a power, raised to another power, is the base raised to the first power times the second power.
Negative Exponent Rule $a^{- n} = \frac{1}{a^{n}}$ provided that $a not equal to 0$	$6^{- 8} = \frac{1}{6^{8}}$ $\frac{1}{12^{7}} = 12^{- 7}$	A base raised to a negative exponent is 1 divided by the base raised to the positive power, and vice versa.

These rules often occur in tandem with each other, but it requires that you carefully apply the rules.

Example 10 Simplifying Expressions Using Exponent Rules

Simplify the following:

${(\frac{4^{2} \times 7}{9^{3}})}^{5}$
${(\frac{5 a^{4}}{b^{9}})}^{6}$

Step 1: To simplify this, we start by distributing the power 5 across the quotient:
${(\frac{4^{2} \times 7}{9^{3}})}^{5} = \frac{{(4^{2} \times 7)}^{5}}{{(9^{3})}^{5}}$
Step 2: We distribute the power 5 in the numerator across that multiplication:
${(\frac{4^{2} \times 7}{9^{3}})}^{5} = {\frac{(4^{2} \times 7)}{{(9^{3})}^{5}}}^{5} = \frac{{(4^{2})}^{5} \times 7^{5}}{{(9^{3})}^{5}}$
Step 3: We apply the power rule where indicated:
${(\frac{4^{2} \times 7}{9^{3}})}^{5} = {\frac{(4^{2} \times 7)}{{(9^{3})}^{5}}}^{5} = \frac{{(4^{2})}^{5} \times 7^{5}}{{(9^{3})}^{5}} = \frac{4^{(2 \times 5)} \times 7^{5}}{9^{(3 \times 5)}} = \frac{4^{10} \times 7^{5}}{9^{15}}$
Step 1: To simplify this, we start by distributing the power 6 across the quotient:
${(\frac{5 a^{4}}{b^{9}})}^{9} = \frac{{(5 \times a^{4})}^{6}}{{(b^{9})}^{6}}$
Step 2: We distribute the power 5 in the numerator across that multiplication:
$\frac{{(5 \times a^{4})}^{6}}{{(b^{9})}^{6}} = \frac{{(5)}^{6} \times {(a^{4})}^{6}}{{(b^{9})}^{6}}$
Step 3: We apply the power rule where indicated:
$\frac{{(5)}^{6} \times {(a^{4})}^{6}}{{(b^{9})}^{6}} = \frac{5^{6} a^{24}}{b^{54}}$

Key Terms

base
exponent

Key Concepts

Exponents are used to express multiplying a number by itself a number of times. The number being multiplied by itself is the base. The number of times it is multiplied by itself is the exponent, which is often referred to as the power.
Understanding that exponents represent repeated multiplication of a base makes it possible to establish some rules for combining exponential expressions, using the product rule, the quotient rule, and the power rule. Additionally, it allows us to formulate distributive rules for exponents.
Any non-zero number raised to the 0th power is 1. This makes the definition of the 0th power consistent with the division rule for exponents.
For consistency, negative exponents represent the reciprocal of the base raised to the power, so that $a^{- n} = \frac{1}{a^{n}}$ , provided that $a not equal to 0$ .

Formulas

$a^{n} a^{m} = a^{n + m}$
$\frac{a^{n}}{a^{m}} = a^{(n - m)}$
$a^{0} = 1$ , provided that $a not equal to 0$
${(a \times b)}^{n} = a^{n} \times b^{n}$
${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$
${(a^{n})}^{m} = a^{(n \times m)}$
$a^{- n} = \frac{1}{a^{n}}$ , provided that $a not equal to 0$

Videos

Adapted from Contemporary Mathematics by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.