2.5: Properties of Multiplication

Last updated
Save as PDF

Page ID: 48840

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

Properties of Multiplication

understand and appreciate the commutative and associative properties of multiplication
understand why 1 is the multiplicative identity

The Commutative Property of Multiplication

Commutative Property of Multiplication
The product of two whole numbers is the same regardless of the order of the factors.

Sample Set A

Multiply the two whole numbers.

$6 and 7.$

Solution

$6 \cdot 7 = 42$
$7 \cdot 6 = 42$

The numbers 6 and 7 can be multiplied in any order. Regardless of the order they are multiplied, the product is 42.

Practice Set A

Use the commutative property of multiplication to find the products in two ways.

$15 and 6.$

Answer: $15 \cdot 6 = 90$ and $6 \cdot 15 = 90$

Practice Set A

$432 and 428.$

Answer: $432 \cdot 428 = 184,896$ and $428 \cdot 432 = 184,896$

The Associative Property of Multiplication

Associative Property of Multiplication
If three whole numbers are multiplied, the product will be the same if the first two are multiplied first and then that product is multiplied by the third, or if the second two are multiplied first and that product is multiplied by the first. Note that the order of the factors is maintained.

It is a common mathematical practice to use parentheses to show which pair of numbers is to be combined first.

Sample Set B

Multiply the whole numbers.

$8, 3, and 14.$

Solution

$(8 \cdot 3) \cdot 14 = 24 \cdot 14 = 336$
$8 \cdot (3 \cdot 14) = 8 \cdot 42 = 336$

Practice Set B

Use the associative property of multiplication to find the products in two ways.

$7, 3 and 8.$

Answer: 168

Practice Set B

$73, 18, and 126.$

Answer: 165,564

The Multiplicative Identity

Definition: The Multiplication Identity is 1

The whole number 1 is called the multiplicative identity, since any whole number multiplied by 1 is not changed.

Sample Set C

Multiply the whole numbers.

$12 and 1.$

Solution

$12 \cdot 1 = 12$
$1 \cdot 12 = 12$

Practice Set C

Multiply the whole numbers.

$843 and 1.$

Answer: 843

Exercises

For the following problems, multiply the numbers.

Exercise $\PageIndex{1}$

$9 and 26.$

Answer: 234

Exercise $\PageIndex{2}$

$18 and 41.$

Exercise $\PageIndex{3}$

$96 and 42.$

Answer: 4,032

Exercise $\PageIndex{4}$

$132 and 6.$

Exercise $\PageIndex{5}$

$1000 and 326.$

Answer: 326,000

Exercise $\PageIndex{6}$

$70 and 1400.$

Exercise $\PageIndex{7}$

$3, 12, and 7.$

Answer: 252

Exercise $\PageIndex{8}$

$16, 40, and 5.$

Exercise $\PageIndex{9}$

$10, 97, and 22.$

Answer: 21,340

Exercise $\PageIndex{10}$

$110, 0, and 85.$

Exercise $\PageIndex{11}$

$1, 462, and 18.$

Answer: 8,316

Exercise $\PageIndex{12}$

$3,178, 101, and 5.$

For the following 4 problems, show that the quantities yield the same products by performing the multiplications.

Exercise $\PageIndex{13}$

$(4 \cdot 8) \cdot 2$ and $4 \cdot (8 \cdot 2)$

Answer: $32 \cdot 2 = 64 = 4 \cdot 16$

Exercise $\PageIndex{14}$

$(100 \cdot 62) \cdot 4$ and $100 \cdot (62 \cdot 4)$

Exercise $\PageIndex{15}$

$23 \cdot (11 \cdot 106)$ and $(23 \cdot 11) \cdot 106$

Answer: $23 \cdot 1,166 = 26,818 = 253 \cdot 106$

Exercise $\PageIndex{16}$

$1 \cdot (5 \cdot 2)$ and $(1 \cdot 5) \cdot 2$

Exercise $\PageIndex{17}$

The fact that

$(\text{a first number } \cdot \text{ a second number}) \cdot \text{a third number} = \text{a first number} \cdot (\text{a second number } \cdot \text{ a third number})$

an example of the property of multiplication.

Answer: associative

Exercise $\PageIndex{18}$

The fact that $1 \cdot \text{ any number} = \text{that particular number}$ is an example of the property of multiplication.

Exercise $\PageIndex{19}$

Use the numbers 7 and 9 to illustrate the commutative property of multiplication.

Answer: $7 \cdot 9 = 63 = 9 \cdot 7$

Exercise $\PageIndex{20}$

Use the numbers 6, 4, and 7 to illustrate the associative property of multiplication.

Exercises for Review

Exercise $\PageIndex{21}$

In the number 84,526,098,441, how many millions are there?

Answer: 6

Exercise $\PageIndex{22}$

Replace the letter m with the whole number that makes the addition true.

\[\begin{array} {r} {85} \\ {\underline{+\ \ \ m}} \\ {97} \end{array}\nonumber\]

Exercise $\PageIndex{23}$

Use the numbers 4 and 15 to illustrate the commutative property of addition.

Answer: $4 + 15 = 19$
$15 + 4 = 19$

Exercise $\PageIndex{24}$

Find the product. $8,000,000 \times 1,000$

Exercise $\PageIndex{25}$

Specify which of the digits 2, 3, 4, 5, 6, 8,10 are divisors of the number 2.

Answer: 2, 3, 4, 6