Use Multiplication Properties of Exponents

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 $2 \cdot 2 \cdot 2 \cdot 2 .$ This format is known as exponential notation.

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

On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.

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

Simplify:

ⓐ $5^{3}$
ⓑ $9^{1}$

Solution

This table details the step-by-step process of evaluating the exponential expression 5^3, illustrating its expansion and final simplification.
ⓐ
	$5^{3}$
Multiply 3 factors of 5.	$5 \cdot 5 \cdot 5$
Simplify.	$125$

This table illustrates the definition and calculation of 9 raised to the power of 1, showing the expression and its numerical value.
ⓑ
	$9^{1}$
Multiply 1 factor of 9.	$9$

Simplify:

ⓐ ${(\frac{7}{8})}^{2}$
ⓑ ${(0.74)}^{2}$

Solution

Steps illustrating the evaluation of the square of a fraction (7/8) through multiplication and simplification.
ⓐ
	${(\frac{7}{8})}^{2}$
Multiply two factors.	$(\frac{7}{8}) (\frac{7}{8})$
Simplify.	$\frac{49}{64}$

Illustrates the steps to calculate the square of 0.74, from the initial power expression to the final simplified decimal result.
ⓑ
	${(0.74)}^{2}$
Multiply two factors.	$(0.74) (0.74)$
Simplify.	$0.5476$

Simplify:

ⓐ ${(−3)}^{4}$
ⓑ ${−3}^{4}$

Solution

Step-by-step evaluation of the exponential expression (-3)^4.
ⓐ
	${(−3)}^{4}$
Multiply four factors of −3.	$(−3) (−3) (−3) (−3)$
Simplify.	$81$

Step-by-step simplification of the mathematical expression -3^4, demonstrating the calculation process and final result.
ⓑ
	${−3}^{4}$
Multiply two factors.	$- (3 \cdot 3 \cdot 3 \cdot 3)$
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 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.
We write:	$x^{2} \cdot x^{3}$ $x^{2 + 3}$ $x^{5}$

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

An example with numbers helps to verify this property.

\begin{matrix} 2^{2} \cdot 2^{3} & \overset{?}{=} & 2^{2 + 3} \\ 4 \cdot 8 & \overset{?}{=} & 2^{5} \\ 32 & = & 32 ✓ \end{matrix}

Simplify: $x^{5} \cdot x^{7} .$

Solution

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

Simplify: $b^{4} \cdot b .$

Solution

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

Simplify: $2^{7} \cdot 2^{9} .$

Solution

	$2^{7} \cdot 2^{9}$
Use the product property, $a^{m} \cdot a^{n} = a^{m + n} .$
Simplify.	$2^{16}$

Simplify: $y^{17} \cdot y^{23} .$

Solution

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

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

Simplify: $x^{3} \cdot x^{4} \cdot x^{2} .$

Solution

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

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.



What does this mean? How many factors altogether?
So, we have
Notice that 6 is the product of the exponents, 2 and 3.
We write:	${(x^{2})}^{3}$ $x^{2 \cdot 3}$ $x^{6}$

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

An example with numbers helps to verify this property.

\begin{matrix} {(5^{2})}^{3} & \overset{?}{=} & 5^{2 \cdot 3} \\ {(25)}^{3} & \overset{?}{=} & 5^{6} \\ 15,625 & = & 15,625 ✓ \end{matrix}

Simplify:

ⓐ ${(x^{5})}^{7}$
ⓐ ${(3^{6})}^{8}$

Solution

ⓐ
	${(x^{5})}^{7}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$
Simplify.	$x^{35}$

ⓑ
	${(3^{6})}^{8}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$
Simplify.	$3^{48}$

Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Look for a pattern.

This table demonstrates the Power of a Product Rule by showing the step-by-step expansion and simplification of (2x)^3 into 2^3 * x^3.
	${(2 x)}^{3}$
What does this mean?	$2 x \cdot 2 x \cdot 2 x$
We group the like factors together.	$2 \cdot 2 \cdot 2 \cdot x \cdot x \cdot x$
How many factors of 2 and of $x ?$	$2^{3} \cdot x^{3}$
Notice that each factor was raised to the power.	${(2 x)}^{3} is 2^{3} \cdot x^{3}$
We write:	${(2 x)}^{3}$ $2^{3} \cdot x^{3}$

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

An example with numbers helps to verify this property:

\begin{matrix} {(2 \cdot 3)}^{2} & \overset{?}{=} & 2^{2} \cdot 3^{2} \\ 6^{2} & \overset{?}{=} & 4 \cdot 9 \\ 36 & = & 36 ✓ \end{matrix}

Simplify: ${(−11 x)}^{2} .$

Solution

	${(−11 x)}^{2}$
Use the Power of a Product Property, ${(a b)}^{m} = a^{m} b^{m} .$
Simplify.	$121 x^{2}$

Simplify: ${(3 x y)}^{3} .$

Solution

	${(3 x y)}^{3}$
Raise each factor to the third power.
Simplify.	$27 x^{3} y^{3}$

Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

Simplify: ${(x^{2})}^{6} {(x^{5})}^{4} .$

Solution

Step-by-step simplification of an exponential expression, applying the power property and adding exponents.
	${(x^{2})}^{6} {(x^{5})}^{4}$
Use the Power Property.	$x^{12} \cdot x^{20}$
Add the exponents.	$x^{32}$

Simplify: ${(- 7 x^{3} y^{4})}^{2} .$

Solution

This table demonstrates the step-by-step simplification of the algebraic expression (-7x^3y^4)^2 using the Power Property of exponents.
	${(- 7 x^{3} y^{4})}^{2}$
Take each factor to the second power.	${(−7)}^{2} {(x^{3})}^{2} {(y^{4})}^{2}$
Use the Power Property.	$49 x^{6} y^{8}$

Simplify: ${(6 n)}^{2} (4 n^{3}) .$

Solution

Step-by-step simplification of an algebraic expression, demonstrating exponent rules and properties.
	${(6 n)}^{2} (4 n^{3})$
Raise $6 n$ to the second power.	$6^{2} n^{2} \cdot 4 n^{3}$
Simplify.	$36 n^{2} \cdot 4 n^{3}$
Use the Commutative Property.	$36 \cdot 4 \cdot n^{2} \cdot n^{3}$
Multiply the constants and add the exponents.	$144 n^{5}$

Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the n.

Simplify: ${(3 p^{2} q)}^{4} {(2 p q^{2})}^{3} .$

Solution

Step-by-step simplification of the algebraic expression (3p^2q)^4(2pq^2)^3 to its simplified form, demonstrating exponent properties.
	${(3 p^{2} q)}^{4} {(2 p q^{2})}^{3}$
Use the Power of a Product Property.	$3^{4} {(p^{2})}^{4} q^{4} \cdot 2^{3} p^{3} {(q^{2})}^{3}$
Use the Power Property.	$81 p^{8} q^{4} \cdot 8 p^{3} q^{6}$
Use the Commutative Property.	$81 \cdot 8 \cdot p^{8} \cdot p^{3} \cdot q^{4} \cdot q^{6}$
Multiply the constants and add the exponents for each variable.	$648 p^{11} q^{10}$

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

Multiply: $(4 x^{2}) (- 5 x^{3}) .$

Solution

Steps to multiply two monomials, illustrating the rearrangement of factors using the Commutative Property to simplify the expression.
	$(4 x^{2}) (- 5 x^{3})$
Use the Commutative Property to rearrange the factors.	$4 \cdot (−5) \cdot x^{2} \cdot x^{3}$
Multiply.	$- 20 x^{5}$

Multiply: $(\frac{3}{4} c^{3} d) (12 c d^{2}) .$

Solution

Step-by-step simplification of an algebraic expression, showing the application of the Commutative Property to rearrange factors before multiplication.
	$(\frac{3}{4} c^{3} d) (12 c d^{2})$
Use the Commutative Property to rearrange the factors.	$\frac{3}{4} \cdot 12 \cdot c^{3} \cdot c \cdot d \cdot d^{2}$
Multiply.	$9 c^{4} d^{3}$

Key Concepts

Exponential Notation
This is read $a$ to the $m^{th}$ power.
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.
Power Property for Exponents
- If $a$ is a real number and $m, n$ are counting numbers, then
  ${(a^{m})}^{n} = a^{m \cdot n}$
Product to a Power Property for Exponents
- If $a$ and $b$ are real numbers and $m$ is a whole number, then
  ${(a b)}^{m} = a^{m} b^{m}$

Practice Makes Perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

$4^{5}$

Solution

1,024

$10^{3}$

${(\frac{1}{2})}^{2}$

Solution

$\frac{1}{4}$

${(\frac{3}{5})}^{2}$

${(0.2)}^{3}$

Solution

0.008

${(0.4)}^{3}$

${(−5)}^{4}$

Solution

625

${(−3)}^{5}$

${−5}^{4}$

Solution

−625

${−3}^{5}$

${−10}^{4}$

Solution

−10,000

${−2}^{6}$

${(- \frac{2}{3})}^{3}$

Solution

$- \frac{8}{27}$

${(- \frac{1}{4})}^{4}$

$- {0.5}^{2}$

Solution

−0.25

$- {0.1}^{4}$

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression using the Product Property of Exponents.

$x^{3} \cdot x^{6}$

Solution

x⁹

$m^{4} \cdot m^{2}$

$a \cdot a^{4}$

Solution

a⁵

$y^{12} \cdot y$

$3^{5} \cdot 3^{9}$

Solution

3¹⁴

$5^{10} \cdot 5^{6}$

$z \cdot z^{2} \cdot z^{3}$

Solution

z⁶

$a \cdot a^{3} \cdot a^{5}$

$x^{a} \cdot x^{2}$

Solution

x^a+2

$y^{p} \cdot y^{3}$

$y^{a} \cdot y^{b}$

Solution

y^a+b

$x^{p} \cdot x^{q}$

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression using the Power Property of Exponents.

${(u^{4})}^{2}$

Solution

u⁸

${(x^{2})}^{7}$

${(y^{5})}^{4}$

Solution

y²⁰

${(a^{3})}^{2}$

${(10^{2})}^{6}$

Solution

10¹²

${(2^{8})}^{3}$

${(x^{15})}^{6}$

Solution

x⁹⁰

${(y^{12})}^{8}$

${(x^{2})}^{y}$

Solution

x^2y

${(y^{3})}^{x}$

${(5^{x})}^{y}$

Solution

5^xy

${(7^{a})}^{b}$

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

${(5 a)}^{2}$

Solution

25a²

${(7 x)}^{2}$

${(- 6 m)}^{3}$

Solution

−216m³

${(- 9 n)}^{3}$

${(4 r s)}^{2}$

Solution

16r²s²

${(5 a b)}^{3}$

${(4 x y z)}^{4}$

Solution

256x⁴y⁴z⁴

${(- 5 a b c)}^{3}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

${(x^{2})}^{4} \cdot {(x^{3})}^{2}$

Solution

x¹⁴

${(y^{4})}^{3} \cdot {(y^{5})}^{2}$

${(a^{2})}^{6} \cdot {(a^{3})}^{8}$

Solution

a³⁶

${(b^{7})}^{5} \cdot {(b^{2})}^{6}$

${(3 x)}^{2} (5 x)$

Solution

45x³

${(2 y)}^{3} (6 y)$

${(5 a)}^{2} {(2 a)}^{3}$

Solution

200a⁵

${(4 b)}^{2} {(3 b)}^{3}$

${(2 m^{6})}^{3}$

Solution

8m¹⁸

${(3 y^{2})}^{4}$

${(10 x^{2} y)}^{3}$

Solution

1,000x⁶y³

${(2 m n^{4})}^{5}$

${(−2 a^{3} b^{2})}^{4}$

Solution

16a¹²b⁸

${(−10 u^{2} v^{4})}^{3}$

${(\frac{2}{3} x^{2} y)}^{3}$

Solution

$\frac{8}{27} x^{6} y^{3}$

${(\frac{7}{9} p q^{4})}^{2}$

${(8 a^{3})}^{2} {(2 a)}^{4}$

Solution

1,024a¹⁰

${(5 r^{2})}^{3} {(3 r)}^{2}$

${(10 p^{4})}^{3} {(5 p^{6})}^{2}$

Solution

25,000p²⁴

${(4 x^{3})}^{3} {(2 x^{5})}^{4}$

${(\frac{1}{2} x^{2} y^{3})}^{4} {(4 x^{5} y^{3})}^{2}$

Solution

x¹⁸y¹⁸

${(\frac{1}{3} m^{3} n^{2})}^{4} {(9 m^{8} n^{3})}^{2}$

${(3 m^{2} n)}^{2} {(2 m n^{5})}^{4}$

Solution

144m⁸n²²

${(2 p q^{4})}^{3} {(5 p^{6} q)}^{2}$

Multiply Monomials

In the following exercises, multiply the following monomials.

$(12 x^{2}) (−5 x^{4})$

Solution

−60x⁶

$(−10 y^{3}) (7 y^{2})$

$(−8 u^{6}) (−9 u)$

Solution

72u⁷

$(−6 c^{4}) (−12 c)$

$(\frac{1}{5} r^{8}) (20 r^{3})$

Solution

4r¹¹

$(\frac{1}{4} a^{5}) (36 a^{2})$

$(4 a^{3} b) (9 a^{2} b^{6})$

Solution

36a⁵b⁷

$(6 m^{4} n^{3}) (7 m n^{5})$

$(\frac{4}{7} x y^{2}) (14 x y^{3})$

Solution

8x²y⁵

$(\frac{5}{8} u^{3} v) (24 u^{5} v)$

$(\frac{2}{3} x^{2} y) (\frac{3}{4} x y^{2})$

Solution

$\frac{1}{2} x^{3} y^{3}$

$(\frac{3}{5} m^{3} n^{2}) (\frac{5}{9} m^{2} n^{3})$

Everyday Math

Email Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is $6^{2},$ on the third round is $6^{3},$ as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.

Round	Number of people
$1$	$6$
$2$	$6^{2}$
$3$	$6^{3}$
$\dots$	$\dots$
$8$	$?$

Solution

1,679,616

Salary Raul’s boss gives him a $5%$ raise every year on his birthday. This means that each year, Raul’s salary is $1.05$ times his last year’s salary. If his original salary was $$40,000$ , his salary after $1$ year was $$40,000 (1.05),$ after $2$ years was $$40,000 {(1.05)}^{2},$ after $3$ years was $$40,000 {(1.05)}^{3},$ as shown in the table below. What will Raul’s salary be after $10$ years? Simplify the expression, to show Raul’s salary in dollars.

Year	Salary
$1$	$$40,000 (1.05)$
$2$	$$40,000 {(1.05)}^{2}$
$3$	$$40,000 {(1.05)}^{3}$
$\dots$	$\dots$
$10$	$?$

Writing Exercises

Use the Product Property for Exponents to explain why $x \cdot x = x^{2} .$

Solution

Answers will vary.

Explain why ${−5}^{3} = {(−5)}^{3}$ but ${−5}^{4} \neq {(−5)}^{4} .$

Jorge thinks ${(\frac{1}{2})}^{2}$ is $1 .$ What is wrong with his reasoning?

Solution

Answers will vary.

Explain why $x^{3} \cdot x^{5}$ is $x^{8},$ and not $x^{15} .$

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment checklist for students to rate their understanding of simplifying expressions with exponents and multiplying monomials, using categories like 'Confidently' and 'No-I don't get it!'

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?