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Mathematics LibreTexts

10.2: Use Multiplication Properties of Exponents (Part 1)

 

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Skills to Develop

  • Simplify expressions with exponents
  • Simplify expressions using the Product Property of Exponents
  • Simplify expressions using the Power Property of Exponents
  • Simplify expressions using the Product to a Power Property
  • Simplify expressions by applying several properties
  • Multiply monomials

be prepared!

Before you get started, take this readiness quiz.

  1. Simplify: \(\frac{3}{4} \cdot \frac{3}{4}\). If you missed the problem, review Example 4.25.
  2. Simplify: (−2)(−2)(−2). If you missed the problem, review Example 3.52.

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, 24 means to multiply four factors of 2, so 24 means 2 • 2 • 2 • 2. This format is known as exponential notation.

Definition: Exponential Notation

CNX_BMath_Figure_10_02_013_img.jpg

This is read a to the mth power.

In the expression am, the exponent tells us how many times we use the base a as a factor.

CNX_BMath_Figure_10_02_014_img.jpg

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

Example 10.11:

Simplify: (a) 53 (b) 91

Solution

(a) 53

Multiply 3 factors of 5. 5 • 5 • 5
Simplify. 125

(b) 91

Multiply 1 factor of 9. 9

Exercise 10.21:

Simplify: (a) 43 (b) 111

Exercise 10.22:

Simplify: (a) 34 (b) 211

Example 10.12:

Simplify: (a) \(\left(\dfrac{7}{8}\right)^{2}\) (b) (0.74)2

Solution

(a) \(\left(\dfrac{7}{8}\right)^{2}\)

Multiply two factors. $$\left(\dfrac{7}{8}\right) \left(\dfrac{7}{8}\right)$$
Simplify. $$\frac{49}{64}$$

(b) (0.74)2

Multiply two factors. (0.74)(0.74)
Simplify. 0.5476

Exercise 10.23:

Simplify: (a) \(\left(\dfrac{5}{8}\right)^{2}\) (b) (0.67)2

Exercise 10.24:

Simplify: (a) \(\left(\dfrac{2}{5}\right)^{3}\) (b) (0.127)2

Example 10.13:

Simplify: (a) (−3)4 (b) −34

Solution

(a) (−3)4

Multiply four factors of −3. (−3)(−3)(−3)(−3)
Simplify. 81

(b) −34

Multiply two factors. −(3 • 3 • 3 • 3)
Simplify. −81

Notice the similarities and differences in parts (a) and (b). Why are the answers different? In part (a) the parentheses tell us to raise the (−3) to the 4th power. In part (b) we raise only the 3 to the 4th power and then find the opposite.

Exercise 10.25:

Simplify: (a) (−2)4 (b) −24

Exercise10.26:

Simplify: (a) (−8)2 (b) −82

Simplify Expressions Using the Product Property of Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.

First, we will look at an example that leads to the Product Property.

  $$x^{2} \cdot x^{2}$$
What does this mean? How many factors altogether? CNX_BMath_Figure_10_02_015_img-02.png
So, we have $$x^{5}$$
Notice that 5 is the sum of the exponents, 2 and 3. $$x^{2} \cdot x^{3}\; is\; x^{2+3},\; or\; x^{5}$$
We write: $$\begin{split} &x^{2} \cdot x^{3} \\ &x^{2+3} \\ &x^{5} \end{split}$$

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Definition: Product Property of Exponents

If a is a real number and m, n are counting numbers, then

$$a^{m} \cdot a^{n} = a^{m + n}$$

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

$$\begin{split} 2^{2} \cdot 2^{3} &\stackrel{?}{=} 2^{2+3} \\ 4 \cdot 8 &\stackrel{?}{=} 2^{5} \\ 32 &= 32\; \checkmark \end{split}$$

Example 10.14:

Simplify: x5 • x7.

Solution
Use the product property, am • an = am + n. $$x^{\textcolor{red}{5+7}}$$
Simplify. $$x^{12}$$

Exercise 10.27:

Simplify: x7 • x8.

Exercise 10.28:

Simplify: x5 • x11.

Example 10.15:

Simplify: b4 • b.

Solution
Rewrite, b = b1. $$b^{4} \cdot b^{1}$$
Use the product property, am • an = am + n. $$b^{\textcolor{red}{4+1}}$$
Simplify. $$b^{5}$$

Exercise 10.29:

Simplify: p9 • p.

Exercise 10.30:

Simplify: m • m7.

Example 10.16:

Simplify: 27 • 29.

Solution
Use the product property, am • an = am + n. $$2^{\textcolor{red}{7+9}}$$
Simplify. $$2^{16}$$

Exercise 10.31:

Simplify: 6 • 69.

Exercise 10.32:

Simplify: 96 • 99.

Example 10.17:

Simplify: y17 • y23.

Solution
Notice, the bases are the same, so add the exponents. $$y^{\textcolor{red}{17+23}}$$
Simplify. $$y^{40}$$

Exercise 10.33:

Simplify: y24 • y19.

Exercise 10.34:

Simplify: z15 • z24.

We can extend the Product Property of Exponents to more than two factors.

Example 10.18:

Simplify: x3 • x4 • x2.

Solution
Add the exponents, since the bases are the same. $$x^{\textcolor{red}{3+4+2}}$$
Simplify. $$x^{9}$$

Exercise 10.35:

Simplify: x7 • x5 • x9.

Exercise 10.36:

Simplify: y3 • y8 • y4.

Simplify Expressions Using the Power Property of Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

  $$(x^{2})^{3}$$
What does this mean? $$x^{2} \cdot x^{2} \cdot x^{2}$$
How many factors altogether? CNX_BMath_Figure_10_02_021_img-03.png
So, we have $$x^{6}$$
Notice that 6 is the product of the exponents, 2 and 3. $$(x^{2})^{3}\; is\; x^{2 \cdot 3}\; or\; x^{6}$$
We write: $$\begin{split} &(x^{2})^{3} \\ &x^{2 \cdot 3} \\ &x^{6} \end{split}$$

We multiplied the exponents. This leads to the Power Property for Exponents.

Definition: Power Property of Exponents

If a is a real number and m, n are whole numbers, then

$$(a^{m})^{n} = a^{m \cdot n}$$

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

$$\begin{split} (5^{2})^{3} &\stackrel{?}{=} 5^{2 \cdot 3} \\ (25)^{3} &\stackrel{?}{=} 5^{6} \\ 15,625 &= 15,625\; \checkmark \end{split}$$

Example 10.19:

Simplify: (a) (x5)7 (b) (36)8

Solution

(a) (x5)7

Use the Power Property, (am)n = am • n. $$x^{\textcolor{red}{5 \cdot 7}}$$
Simplify. $$x^{35}$$

(b) (36)8

Use the Power Property, (am)n = am • n. $$3^{\textcolor{red}{6 \cdot 8}}$$
Simplify. $$x^{48}$$

Exercise 10.37:

Simplify: (a) (x7)4 (b) (74)8

Exercise 10.38:

Simplify: (a) (x6)9 (b) (86)7

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