5.1 Use Multiplication Properties of Exponents – Intermediate Algebra II (2023)

Learning Objectives

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

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

Remember that an exponent indicates repeated multiplication of the same quantity. For example, means to multiply 2 by itself 4 times, so means 2 · 2 · 2 · 2

Let’s review the vocabulary for expressions with exponents.

Exponential Notation

This is read to the power.

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

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

EXAMPLE 1

Simplify: a) b) c) d) .

Solution

a)
Multiply three factors of 4.	4 · 4 · 4
Simplify.
b)
Multiply one factor of 7.
c)
Multiply two factors.
Simplify.
d)
Multiply two factors.
Simplify.

TRY IT 1.1

Simplify: a) b) c) d) .

Show answer

a) 216 b) c) d) 0.1849

TRY IT 1.2

Simplify: a) b) c) d) .

Show answer

a) b) 21 c) d)

EXAMPLE 2

Simplify: a) b) .

Solution

a)
Multiply four factors of .
Simplify.
b)
Multiply four factors of 5.	-(5 · 5 · 5 · 5)
Simplify.

TRY IT 2.1

Simplify: a) b) .

Show answer

a) b)

TRY IT 2.2

Simplify: a) b) .

Show answer

a) b)

Notice the similarities and differences in (Example 2) a) and (Example 2) b)! Why are the answers different? As we follow the order of operations in part a) the parentheses tell us to raise the to the 4^th power. In part b) we raise just the 5 to the 4^th power and then take the opposite.

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.

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:

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

Product Property for Exponents

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

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

EXAMPLE 3

Simplify: .

Solution


Use the product property,.
Simplify.

TRY IT 3.1

Simplify: .

Show answer

TRY IT 3.2

Simplify: .

Show answer

EXAMPLE 4

Simplify: a) b) .

Solution

Use the product property, a^m · aⁿ = a^m+n.
Simplify.
Use the product property, a^m · aⁿ = a^m+n.
Simplify.

TRY IT 4.1

Simplify: a) b) .

Show answer

a) b)

TRY IT 4.2

Simplify: a) b) .

Show answer

a) b)

EXAMPLE 5

Simplify: a) b) .

Solution

Rewrite, a = a¹.
Use the product property, a^m · aⁿ = a^m+n.
Simplify.
Notice, the bases are the same, so add the exponents.
Simplify.

TRY IT 5.1

Simplify: a) b) .

Show answer

a) b)

TRY IT 5.2

Simplify: a) b) .

Show answer

a) b)

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

EXAMPLE 6

Simplify: .

Solution


Add the exponents, since bases are the same.
Simplify.

TRY IT 6.1

Simplify: .

Show answer

TRY IT 6.2

Simplify: .

Show answer

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:

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

Power Property for Exponents

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

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

An example with numbers helps to verify this property.

EXAMPLE 7

Simplify: a) b) .

Solution


Use the power property, (a^m)ⁿ = a^{m · n}.
Simplify.


Use the power property.
Simplify.

TRY IT 7.1

Simplify: a) b) .

Show answer

a) b)

TRY IT 7.2

Simplify: a) b) .

Show answer

a) b)

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?


What does this mean?
We group the like factors together.
How many factors of 2 and of ?

Notice that each factor was raised to the power and is .

We write:

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

Product to a Power Property for Exponents

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

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

EXAMPLE 8

Simplify: a) b) .

Solution

Use Power of a Product Property, (ab)^m = a^mb^m.
Simplify.
Use Power of a Product Property, (ab)^m = a^mb^m.
Simplify.

TRY IT 8.1

Simplify: a) b) .

Show answer

a) b)

TRY IT 8.2

Simplify: a) b) .

Show answer

a) b)

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.

Properties of Exponents

If and are real numbers, and and are whole numbers, then

Product Property
Power Property
Product to a Power

All exponent properties hold true for any real numbers and . Right now, we only use whole number exponents.

EXAMPLE 9

Simplify: a) b) .

Solution

a)
Use the Power Property.
Add the exponents.
b)
Use the Product to a Power Property.
Use the Power Property.
Simplify.

TRY IT 9.1

Simplify: a) b) .

Show answer

a) b)

TRY IT 9.2

Simplify: a) b) .

Show answer

a) b)

EXAMPLE 10

Simplify: a) b) .

Solution

a)
Raise to the second power.
Simplify.
Use the Commutative Property.
Multiply the constants and add the exponents.
b)
Use the Product to a Power Property.
Simplify.
Use the Commutative Property.
Multiply the constants and add the exponents.

TRY IT 10.1

Simplify: a) b) .

Show answer

a) b)

TRY IT 10.2

Simplify: a) b) .

Show answer

a) b)

A term in algebra is a constant or the product of a constant and one or more variables. When it is of the form , where is a constant and is a whole number, it is called a monomial. Some examples of monomial are , and .

Monomials

A monomial is a term of the form , where is a constant and is a positive whole number.

Since a monomial is an algebraic expression,we can use the properties of exponents to multiply monomials.

EXAMPLE 11

Multiply: .

Solution


Use the Commutative Property to rearrange the terms.
Multiply.

TRY IT 11.1

Multiply: .

Show answer

TRY IT 11.2

Multiply: .

Show answer

EXAMPLE 12

Multiply: .

Solution


Use the Commutative Property to rearrange the terms.
Multiply.

TRY IT 12.1

Multiply: .

Show answer

TRY IT 12.2

Multiply: .

Show answer

Additional Online Resources

Multiplication Properties of Exponents

Exponential Notation
Properties of Exponents
- If are real numbers and are whole numbers, then

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

a)
b)
c)
d)

a)
b)

10.

a)
b)

Simplify Expressions Using the Product Property for Exponents

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

11.	12.
13.	14.
15. a) b)	16. a) b)
17. a) b)	18. a) b)
19.	20.
21.	22.
23.	24.
25.	26.

Simplify Expressions Using the Power Property for Exponents

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

27. a) b)	28. a) b)
29. a) b)	30. 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.

31. a) b)	32. a) b)
33. a) b)	34. a) b)

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

35.

a)
b)

36.

a)
b)

37.

a)
b)

38.

a)
b)

39.

a)
b)

40.

a)
b)

41.

a)
b)

42.

a)
b)

43.

a)
b)

44.

a)
b)

45.

a)
b)

Multiply Monomials

In the following exercises, multiply the terms.

46.	47.
48.	49.
50.	51.
52.	53.
54.	55.
56.	57.

Mixed Practice

In the following exercises, simplify each expression.

58.	59.
60.	61.
62.	63.
64.	65.
66.	67.
68.	69.
70.	71.
72.	73.
74.	75.
76.	77.

Everyday Math

78. Email Kate emails a flyer to ten of her friends and tells them to forward it to ten of their friends, who forward it to ten of their friends, and so on. The number of people who receive the email on the second round is , on the third round is , as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.

Round	Number of people
1	10
2
3
…	…
6	?

79. Salary Jamal’s boss gives him a 3% raise every year on his birthday. This means that each year, Jamal’s salary is 1.03 times his last year’s salary. If his original salary was $35,000, his salary after 1 year was , after 2 years was , after 3 years was , as shown in the table below. What will Jamal’s salary be after 10 years? Simplify the expression, to show Jamal’s salary in dollars.

Year	Salary
1
2
3
…	…
10	?

80. Clearance A department store is clearing out merchandise in order to make room for new inventory. The plan is to mark down items by 30% each week. This means that each week the cost of an item is 70% of the previous week’s cost. If the original cost of a sofa was $1,000, the cost for the first week would be and the cost of the item during the second week would be . Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.

Week	Cost
1
2
3
…	…
5	?

81. Depreciation Once a new car is driven away from the dealer, it begins to lose value. Each year, a car loses 10% of its value. This means that each year the value of a car is 90% of the previous year’s value. If a new car was purchased for ?20,000, the value at the end of the first year would be and the value of the car after the end of the second year would be . Complete the table shown below. What will be the value of the car at the end of the eighth year? Simplify the expression, to show the value in dollars.

Week	Cost
1
2
3
4	…
8	?

Writing Exercises

82. Use the Product Property for Exponents to explain why .	83. Explain why but .
84. Jorge thinks is 1. What is wrong with his reasoning?	85. Explain why is , and not

1. a) 243 b) 9 c) d) 0.0016	3. a) 64 b) 14 c) d) 0.49
5. a) 1,296 b)	7. a) b)
9. a) b)	11.
13.	15. a) b)
17. a) b)	19.
21.	23.
25.	27. a) b)
29. a) b)	31. a) b)
33. a) b)	35. a) b)
37. a) b)	39. a) b)
41. a) b)	43. a) b)
45. a) b)	47.
49.	51.
53.	55.
57.	59.
61.	63.
65.	67.
69.	71.
73.	75.
77.	79. $47,037.07
81. $8,609.34	83. and 85. Answers will vary.

This chapter has been adapted from “Use Multiplication Properties of Exponents” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

FAQs

What are the 5 rules of exponents? ›

Product Rule.
Power-to-Power Rule.
Product-to-Power Rule.
Zero Exponent.
Quotient Rule.
Quotient-to-Power Rule.
Negative Exponent.

Get More Info ›

What is the exponent rule in Algebra 2? ›

The Power Rule for Exponents: (a^m)ⁿ = a^m^*ⁿ. To raise a number with an exponent to a power, multiply the exponent times the power. Negative Exponent Rule: x^–ⁿ = 1/xⁿ. Invert the base to change a negative exponent into a positive.

View Details ›

What are the properties of exponents explain with example? ›

The exponent of a number shows how many times the number is multiplied by itself. For example, 2 × 2 × 2 × 2 can be written as 2⁴, as 2 is multiplied by itself 4 times. Here, 2 is called the 'base' and 4 is called the 'exponent' or 'power'.

Tell Me More ›

What are exponents examples? ›

For example, 8 × 8 × 8 can be expressed as 8³ because 8 is multiplied by itself 3 times. Here, 3 is the 'exponent' or 'power' which tells how many times 8 is multiplied by itself, and 8 is the 'base' which represents the number being multiplied.

Know More ›

How do you solve exponential powers? ›

Step 1: Isolate the exponential expression. Step 2: Take the natural log of both sides. Step 3: Use the properties of logs to pull the x out of the exponent. Step 4: Solve for x.

Keep Reading ›

What is the rule for multiplying exponents? ›

When you're multiplying exponents, use the first rule: add powers together when multiplying like bases. 5^2 × 5^6 = ? The bases of the equation stay the same, and the values of the exponents get added together.

What are the 7 rules for exponents? ›

For all examples below, assume that X and Y are nonzero real numbers and a and b are integers.

RULE 1: Zero Property. ...
RULE 2: Negative Property. ...
RULE 3: Product Property. ...
RULE 4: Quotient Property. ...
RULE 5: Power of a Power Property. ...
RULE 6: Power of a Product Property. ...
RULE 7: Power of a Quotient Property.

Nov 29, 2021

Keep Reading ›

How do exponents work in algebra? ›

Exponents, also known as powers, are values that show how many times to multiply a base number by itself. For example, 43 is telling you to multiply four by itself three times. The number being raised by a power is known as the base, while the superscript number above it is the exponent or power.

Get More Info ›

Do you multiply exponents first? ›

A. The order of operations is the order you use to work out math expressions: parentheses, exponents, multiplication, division, addition, subtraction.

What is properties of exponents in math? ›

Product of a Power: When you multiply exponentials with the same base, you add their exponents (or powers). Power to a Power: When you have a power to a power, you multiply the exponents (or powers). Quotient of Powers: When you divide exponentials with the same base, you subtract the exponents (or powers).

Keep Reading ›

How do you write in exponents? ›

To express a number in exponential notation, write it in the form: c × 10n, where c is a number between 1 and 10 (e.g. 1, 2.5, 6.3, 9.8) and n is an integer (e.g. 1, -3, 6, -2). To find n, count the number of places that the decimal point must be moved to give the coefficient, c.

Learn More Now ›

What is an example of exponent in a sentence? ›

exponent

She was a leading exponent of free trade during her political career.
Huxley was an exponent of Darwin's theory of evolution.

View Details ›

What is an exponent math example sentence? ›

Example Sentences

She has become one of America's foremost exponents of the romantic style in interior design. The exponent 3 in 10³ indicates 10 x 10 x 10.

How do you divide exponents in Algebra 2? ›

To divide them, you take the exponent value in the numerator (the top exponent) and subtract the exponent value of the denominator (the bottom exponent).

View Details ›

What are the basic rules of math? ›

The four basic Mathematical rules are addition, subtraction, multiplication, and division.

What is the basic formula for exponential? ›

An exponential function is defined by the formula f(x) = a^x, where the input variable x occurs as an exponent.

Discover More Details ›

Do you always multiply exponents? ›

Only multiply exponents when taking the power of a power, not when you are multiplying terms. Then, you add the exponents.

What are the 4 basic rules of algebra? ›

Ans. The commutative rule of addition, the commutative rule of multiplication, the associative rule of addition, the associative rule of multiplication, and the distributive property of multiplication make up the fundamental rules of the algebra.

Discover More Details ›

What grade math is exponents? ›

Students will first learn about exponents as part of numbers and operations in base ten in 5th grade, and then as part of expressions and equations in 6th grade.

See Details ›

What is the property of powers? ›

The rule for raising a power to a power is called the Power of a Power Property. The Power of a Power Property states that if an exponent is being raised to another exponent, you can multiply the exponents. You can use this property to solve a problem like ( 3 x 2 ) 3 .

Discover More ›

How do you multiply and divide exponents? ›

There are different rules to follow when multiplying exponents and when dividing exponents. If we are multiplying similar bases, we simply add the exponents. If we are dividing, we simply subtract the exponents. If an exponent is outside the parentheses, it is distributed to the inside terms.

Read The Full Story ›

What are the 7 properties of multiplication? ›

What are the Properties of Multiplication of Integers?

Closure property.
Commutative property.
Associative property.
Distributive property.
Multiplication by zero.
Multiplicative identity.

Aug 31, 2020

Read On ›

What is the 3 properties of multiplication? ›

There are three properties of multiplication: commutative, associative, and distributive.

Find Out More ›

What is the multiplication property of 1 examples? ›

The identity property of 1 says that any number multiplied by 1 keeps its identity. In other words, any number multiplied by 1 stays the same. The reason the number stays the same is because multiplying by 1 means we have 1 copy of the number. For example, 32x1=32.

View Details ›

What are the 4 multiplication properties? ›

The properties of multiplication are distributive, commutative, associative, removing a common factor and the neutral element.

Learn More Now ›

What is multiplication property in math? ›

Multiplication property of equality states that if both the sides of an equation are multiplied by the same number, the expressions on the both sides of the equation remain equal to each other.

Get More Info Here ›

What is the power property of exponents? ›

The Power of a Power Property states that if an exponent is being raised to another exponent, you can multiply the exponents. You can use this property to solve a problem like ( 3 x 2 ) 3 .

What is the difference between exponents with and without parentheses? ›

If the base is in parentheses, as in our first case, the exponent affects everything that is inside the parenthesis, that is, the sign and the number. However, if the base is not in parentheses, as in the second case, the exponent affects only the immediate value to the left, that is, only the number, without the sign.

Find Out More ›


Rewrite, a = a¹.
Use the product property, a^m · aⁿ = a^m+n.
Simplify.

58.	59.
60.	61.
62.	63.
64.	65.
66.	67.
68.	69.
70.	71.
72.	73.
74.	75.
76.	77.

58.	59.
60.	61.
62.	63.
64.	65.
66.	67.
68.	69.
70.	71.
72.	73.
74.	75.
76.	77.

5.1 Use Multiplication Properties of Exponents – Intermediate Algebra II (2023)

Simplify Expressions with Exponents

Simplify Expressions Using the Product Property for Exponents

Simplify Expressions Using the Power Property for Exponents

Simplify Expressions Using the Product to a Power Property

Simplify Expressions by Applying Several Properties

Multiply Monomials

Mixed Practice

Everyday Math

Writing Exercises

FAQs

What are the 5 rules of exponents? ›

How do you write in exponents? ›

What are the 4 basic rules of algebra? ›

What is the difference between exponents with and without parentheses? ›

References

58.	59.
60.	61.
62.	63.
64.	65.
66.	67.
68.	69.
70.	71.
72.	73.
74.	75.
76.	77.