10.2 Use multiplication properties of exponents

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By the end of this section, you will be able to:
• 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

Before you get started, take this readiness quiz.

1. Simplify: $\frac{3}{4}·\frac{3}{4}.$
If you missed the problem, review Multiply and Divide Fractions .
2. Simplify: $\left(-2\right)\left(-2\right)\left(-2\right).$
If you missed the problem, review Multiply and Divide Integers .

Simplify expressions with exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, ${2}^{4}$ means to multiply four factors of $2,$ so ${2}^{4}$ means $2·2·2·2.$ This format is known as exponential notation .

Exponential notation

This is read $a$ to the ${m}^{\mathrm{th}}$ power.

In the expression ${a}^{m},$ the exponent tells us how many times we use the base $a$ as a factor.

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

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{5}^{3}$
2. $\phantom{\rule{0.2em}{0ex}}{9}^{1}$

Solution

 ⓐ ${5}^{3}$ Multiply 3 factors of 5. $5·5·5$ Simplify. $125$
 ⓑ ${9}^{1}$ Multiply 1 factor of 9. $9$

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{4}^{3}$
2. $\phantom{\rule{0.2em}{0ex}}{11}^{1}$

1. 64
2. 11

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{3}^{4}$
2. $\phantom{\rule{0.2em}{0ex}}{21}^{1}$

1. 81
2. 21

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{\left(\frac{7}{8}\right)}^{2}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(0.74\right)}^{2}$

Solution

 ⓐ ${\left(\frac{7}{8}\right)}^{2}$ Multiply two factors. $\left(\frac{7}{8}\right)\left(\frac{7}{8}\right)$ Simplify. $\frac{49}{64}$
 ⓑ ${\left(0.74\right)}^{2}$ Multiply two factors. $\left(0.74\right)\left(0.74\right)$ Simplify. $0.5476$

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{\left(\frac{5}{8}\right)}^{2}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(0.67\right)}^{2}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{25}{64}$
2. $\phantom{\rule{0.2em}{0ex}}0.4489$

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{\left(\frac{2}{5}\right)}^{3}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(0.127\right)}^{2}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{8}{125}$
2. $\phantom{\rule{0.2em}{0ex}}0.016129$

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{\left(-3\right)}^{4}$
2. $\phantom{\rule{0.2em}{0ex}}{-3}^{4}$

Solution

 ⓐ ${\left(-3\right)}^{3}$ Multiply four factors of −3. $\left(-3\right)\left(-3\right)\left(-3\right)\left(-3\right)$ Simplify. $81$
 ⓑ ${-3}^{4}$ Multiply two factors. $-\left(3·3·3·3\right)$ Simplify. $-81$

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

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{\left(-2\right)}^{4}$
2. $\phantom{\rule{0.2em}{0ex}}{-2}^{4}$

1. 16
2. −16

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{\left(-8\right)}^{2}$
2. $\phantom{\rule{0.2em}{0ex}}{-8}^{2}$

1. 64
2. −64

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.

 What does this mean? How many factors altogether? So, we have Notice that 5 is the sum of the exponents, 2 and 3.

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

Product property of exponents

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

${a}^{m}·{a}^{n}={a}^{m+n}$

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

$\begin{array}{ccc}\hfill {2}^{2}·{2}^{3}& \stackrel{?}{=}& {2}^{2+3}\hfill \\ \hfill 4·8& \stackrel{?}{=}& {2}^{5}\hfill \\ \hfill 32& =& 32✓\hfill \end{array}$

Simplify: ${x}^{5}·{x}^{7}.$

Solution

 ${x}^{5}·{x}^{7}$ Use the product property, ${a}^{m}·{a}^{n}={a}^{m+n}.$ Simplify. ${x}^{12}$

Simplify: ${x}^{7}·{x}^{8}.$

x 15

Simplify: ${x}^{5}·{x}^{11}.$

x 16

Simplify: ${b}^{4}·b.$

Solution

 ${b}^{4}·b$ Rewrite, $b={b}^{1}.$ ${b}^{4}·{b}^{1}$ Use the product property, ${a}^{m}·{a}^{n}={a}^{m+n}.$ Simplify. ${b}^{5}$

Simplify: ${p}^{9}·p.$

p 10

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