Properties of Real Numbers

Use the Commutative and Associative Properties

Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?

\begin{matrix} 5 + 3 & 3 + 5 \\ 8 & 8 \end{matrix}

5 + 3 = 3 + 5

The results are the same.

As we can see, the order in which we add does not matter!

What about multiplying $5 and 3 ?$

\begin{matrix} 5 \cdot 3 & 3 \cdot 5 \\ 15 & 15 \end{matrix}

5 \cdot 3 = 3 \cdot 5

Again, the results are the same!

The order in which we multiply does not matter!

These examples illustrate the commutative property. When adding or multiplying, changing the order gives the same result.

The commutative property has to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.

What about subtraction? Does order matter when we subtract numbers? Does $7 - 3$ give the same result as $3 - 7 ?$

\begin{matrix} \begin{matrix} 7 - 3 & 3 - 7 \\ 4 & −4 \end{matrix} \\ 4 \neq − 4 \\ 7 - 3 \neq 3 - 7 \end{matrix}

The results are not the same.

Since changing the order of the subtraction did not give the same result, we know that subtraction is not commutative.

Let’s see what happens when we divide two numbers. Is division commutative?

\begin{matrix} \begin{matrix} 12 \div 4 & 4 \div 12 \\ \frac{12}{4} & \frac{4}{12} \\ 3 & \frac{1}{3} \end{matrix} \\ 3 \neq \frac{1}{3} \\ 12 \div 4 \neq 4 \div 12 \end{matrix}

The results are not the same.

Since changing the order of the division did not give the same result, division is not commutative. The commutative properties only apply to addition and multiplication!

Addition and multiplication are commutative.
Subtraction and Division are not commutative.

If you were asked to simplify this expression, how would you do it and what would your answer be?

7 + 8 + 2

Some people would think $7 + 8 is 15$ and then $15 + 2 is 17 .$ Others might start with $8 + 2 makes 10$ and then $7 + 10 makes 17 .$

Either way gives the same result. Remember, we use parentheses as grouping symbols to indicate which operation should be done first.

Demonstrates the associative property of addition using the numbers 7, 8, and 2, showing that different groupings result in the same total.
Add $7 + 8$ . Add.	$\begin{matrix} (7 + 8) + 2 \\ 15 + 2 \\ 17 \end{matrix}$
Add $8 + 2$ . Add.	$\begin{matrix} 7 + (8 + 2) \\ 7 + 10 \\ 17 \end{matrix}$
	$(7 + 8) + 2 = 7 + (8 + 2)$

When adding three numbers, changing the grouping of the numbers gives the same result.

This is true for multiplication, too.

This table demonstrates the associative property of multiplication with fractions, showing how grouping factors differently does not change the product.
Multiply. $5 \cdot \frac{1}{3}$ Multiply.	$\begin{matrix} (5 \cdot \frac{1}{3}) \cdot 3 \\ \frac{5}{3} \cdot 3 \\ 5 \end{matrix}$
Multiply. $\frac{1}{3} \cdot 3$ . Multiply.	$\begin{matrix} 5 \cdot (\frac{1}{3} \cdot 3) \\ 5 \cdot 1 \\ 5 \end{matrix}$
	$(5 \cdot \frac{1}{3}) \cdot 3 = 5 \cdot (\frac{1}{3} \cdot 3)$

When multiplying three numbers, changing the grouping of the numbers gives the same result.

You probably know this, but the terminology may be new to you. These examples illustrate the associative property.

Let’s think again about multiplying $5 \cdot \frac{1}{3} \cdot 3 .$ We got the same result both ways, but which way was easier? Multiplying $\frac{1}{3}$ and $3$ first, as shown above on the right side, eliminates the fraction in the first step. Using the associative property can make the math easier!

The associative property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.

We saw that subtraction and division were not commutative. They are not associative either.

When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the commutative property of addition to write the like terms together.

Simplify: $18 p + 6 q + 15 p + 5 q .$

Solution

Step-by-step simplification of an algebraic expression by combining like terms using the commutative property of addition.
	$18 p + 6 q + 15 p + 5 q$
Use the commutative property of addition to re-order so that like terms are together.	$18 p + 15 p + 6 q + 5 q$
Add like terms.	$33 p + 11 q$

When we have to simplify algebraic expressions, we can often make the work easier by applying the commutative or associative property first, instead of automatically following the order of operations. When adding or subtracting fractions, combine those with a common denominator first.

Simplify: $(\frac{5}{13} + \frac{3}{4}) + \frac{1}{4} .$

Solution

Step-by-step solution simplifying a fractional expression by grouping terms with common denominators for easier calculation.
	$(\frac{5}{13} + \frac{3}{4}) + \frac{1}{4}$
Notice that the last 2 terms have a common denominator, so change the grouping.	$\frac{5}{13} + (\frac{3}{4} + \frac{1}{4})$
Add in parentheses first.	$\frac{5}{13} + (\frac{4}{4})$
Simplify the fraction.	$\frac{5}{13} + 1$
Add.	$1 \frac{5}{13}$
Convert to an improper fraction.	$\frac{18}{13}$

Use the associative property to simplify $6 (3 x) .$

Solution

Steps to simplify the algebraic expression 6(3x) using the associative property of multiplication.
	$6 (3 x)$
Change the grouping.	$(6 \cdot 3) x$
Multiply in the parentheses.	$18 x$

Notice that we can multiply $6 \cdot 3$ but we could not multiply 3x without having a value for x.

Use the Identity and Inverse Properties of Addition and Multiplication

What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity.

For example,

\begin{matrix} 13 + 0 & −14 + 0 & 0 + (−8) \\ 13 & - 14 & - 8 \end{matrix}

These examples illustrate the Identity Property of Addition that states that for any real number $a,$ $a + 0 = a$ and $0 + a = a .$

What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity.

For example,

\begin{matrix} 43 \cdot 1 & - 27 \cdot 1 & 1 \cdot \frac{3}{5} \\ 43 & - 27 & \frac{3}{5} \end{matrix}

These examples illustrate the Identity Property of Multiplication that states that for any real number $a,$ $a \cdot 1 = a$ and $1 \cdot a = a .$

We summarize the Identity Properties below.

In the top line of this figure, we have the question “What number added to 5 gives the additive identity, 0?” On the following line, we have 5 plus a blank space equals 0. Then it is stated that “We know 5 plus negative 5 equals 0.” On the following line, we have the question “What number added to negative 6 gives the additive identity, 0?” On the following line, we have negative 6 plus a blank space equals 0. Then it is stated that “We know negative 6 plus 6 equals 0.”

Notice that in each case, the missing number was the opposite of the number!

We call $− a .$ the additive inverse of a. The opposite of a number is its additive inverse. A number and its opposite add to zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number $a, a + (− a) = 0 .$ Remember, a number and its opposite add to zero.

What number multiplied by $\frac{2}{3}$ gives the multiplicative identity, 1? In other words, $\frac{2}{3}$ times what results in 1?

We have the statement that 2/3 times a blank space equals 1. Then it is stated that “We know 2/3 times 3/2 equals 1.”

What number multiplied by 2 gives the multiplicative identity, 1? In other words 2 times what results in 1?

We have the statement that 2 times a blank space equals 1. Then it is stated that “We know 2 times 1/2 equals 1.”

Notice that in each case, the missing number was the reciprocal of the number!

We call $\frac{1}{a}$ the multiplicative inverse of a. The reciprocal of $a$ number is its multiplicative inverse. A number and its reciprocal multiply to one, which is the multiplicative identity. This leads to the Inverse Property of Multiplication that states that for any real number $a, a \neq 0, a \cdot \frac{1}{a} = 1 .$

We’ll formally state the inverse properties here:

Inverse Property

This table defines additive and multiplicative inverses for real numbers, providing their descriptions and mathematical expressions.
of addition	For any real number $a$ , $− a$ is the additive inverse of $a$ . A number and its opposite add to zero.	$a + (− a) = 0$
of multiplication	For any real number $a$ , $\frac{1}{a}$ is the multiplicative inverse of $a$ . A number and its reciprocal multiply to one.	$a \cdot \frac{1}{a} = 1$

Find the additive inverse of ⓐ $\frac{5}{8}$ ⓑ $0.6$ ⓒ $−8$ ⓓ $- \frac{4}{3} .$

Solution

To find the additive inverse, we find the opposite.

ⓐ The additive inverse of $\frac{5}{8}$ is the opposite of $\frac{5}{8} .$ The additive inverse of $\frac{5}{8}$ is $- \frac{5}{8} .$
ⓑ The additive inverse of 0.6 is the opposite of 0.6. The additive inverse of 0.6 is $−0.6 .$
ⓒ The additive inverse of $−8$ is the opposite of $−8 .$ We write the opposite of $−8$ as $− (−8),$ and then simplify it to 8. Therefore, the additive inverse of $−8$ is 8.
ⓓ The additive inverse of $- \frac{4}{3}$ is the opposite of $- \frac{4}{3} .$ We write this as $− (- \frac{4}{3}),$ and then simplify to $\frac{4}{3} .$ Thus, the additive inverse of $- \frac{4}{3}$ is $\frac{4}{3} .$

Find the multiplicative inverse of ⓐ $9$ ⓑ $- \frac{1}{9}$ ⓒ $0.9 .$

Solution

To find the multiplicative inverse, we find the reciprocal.

ⓐ The multiplicative inverse of 9 is the reciprocal of 9, which is $\frac{1}{9} .$ Therefore, the multiplicative inverse of 9 is $\frac{1}{9} .$
ⓑ The multiplicative inverse of $- \frac{1}{9}$ is the reciprocal of $- \frac{1}{9},$ which is $−9 .$ Thus, the multiplicative inverse of $- \frac{1}{9}$ is $−9 .$
ⓒ To find the multiplicative inverse of 0.9, we first convert 0.9 to a fraction, $\frac{9}{10} .$ Then we find the reciprocal of the fraction. The reciprocal of $\frac{9}{10}$ is $\frac{10}{9} .$ So the multiplicative inverse of 0.9 is $\frac{10}{9} .$

Use the Properties of Zero

The identity property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.

What about division involving zero? What is $0 \div 3 ?$ Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So,

0 \div 3 = 0

We can check division with the related multiplication fact.

12 \div 6 = 2 because 2 \cdot 6 = 12 .

So we know $0 \div 3 = 0$ because $0 \cdot 3 = 0 .$

Now think about dividing by zero. What is the result of dividing 4 by 0? Think about the related multiplication fact: $4 \div 0 = ?$ means $? \cdot 0 = 4 .$ Is there a number that multiplied by 0 gives 4? Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4.

We conclude that there is no answer to $4 \div 0$ and so we say that division by 0 is undefined.

We summarize the properties of zero below.

Properties of Zero

Multiplication by Zero: For any real number a,

This table illustrates the multiplication property of zero, showing that the product of any number and zero is always zero.
$a \cdot 0 = 0 0 \cdot a = 0$	The product of any number and 0 is 0.

Division of Zero, Division by Zero: For any real number $a, a \neq 0$

This table presents mathematical rules for division involving zero, clarifying cases where the result is zero or undefined, along with their verbal descriptions.
$\frac{0}{a} = 0$	Zero divided by any real number except itself is zero.
$\frac{a}{0} is undefined$	Division by zero is undefined.

Simplify: ⓐ $−8 \cdot 0$ ⓑ $\frac{0}{−2}$ ⓒ $\frac{−32}{0} .$

Solution

This table illustrates fundamental mathematical rules for operations involving zero, specifically multiplication by zero and division by zero, with corresponding examples.
ⓐ The product of any real number and 0 is 0.	$\begin{matrix} −8 \cdot 0 \\ 0 \end{matrix}$
ⓑ The product of any real number and 0 is 0.	$\begin{matrix} \frac{0}{−2} \\ 0 \end{matrix}$
ⓒ Division by 0 is undefined.	$\begin{matrix} \frac{−32}{0} \\ Undefined \end{matrix}$

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

Simplify: ⓐ $\frac{0}{n + 5},$ where $n \neq − 5$ ⓑ $\frac{10 - 3 p}{0},$ where $10 - 3 p \neq 0 .$

Solution

Fundamental rules of division involving zero: Zero divided by a non-zero number is zero; division by zero is undefined, shown with mathematical examples.
ⓐ Zero divided by any real number except itself is 0.	$\begin{matrix} \frac{0}{n + 5} \\ 0 \end{matrix}$
ⓑ Division by 0 is undefined.	$\begin{matrix} \frac{10 - 3 p}{0} \\ Undefined \end{matrix}$

Simplify: $−84 n + (−73 n) + 84 n .$

Solution

Step-by-step simplification of the algebraic expression -84n + (-73n) + 84n using the commutative property to combine like terms.
	$−84 n + (−73 n) + 84 n$
Notice that the first and third terms are opposites; use the commutative property of addition to re-order the terms.	$−84 n + 84 n + (−73 n)$
Add left to right.	$0 + (−73 n)$
Add.	$−73 n$

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1.

Simplify: $\frac{7}{15} \cdot \frac{8}{23} \cdot \frac{15}{7} .$

Solution

Demonstrates simplifying a fractional multiplication by reordering factors using the commutative property.
	$\frac{7}{15} \cdot \frac{8}{23} \cdot \frac{15}{7}$
Notice that the first and third terms are reciprocals, so use the commutative property of multiplication to re-order the factors.	$\frac{7}{15} \cdot \frac{15}{7} \cdot \frac{8}{23}$
Multiply left to right.	$1 \cdot \frac{8}{23}$
Multiply.	$\frac{8}{23}$

Simplify: $\frac{3}{4} \cdot \frac{4}{3} (6 x + 12) .$

Solution

This table illustrates the step-by-step simplification of a mathematical expression using reciprocals and the multiplicative identity.
	$\frac{3}{4} \cdot \frac{4}{3} (6 x + 12)$
There is nothing to do in the parentheses, so multiply the two fractions first—notice, they are reciprocals.	$1 (6 x + 12)$
Simplify by recognizing the multiplicative identity.	$6 x + 12$

Simplify Expressions Using the Distributive Property

Suppose that three friends are going to the movies. They each need $9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?

You can think about the dollars separately from the quarters. They need 3 times $9 so $27, and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the distributive property.

Back to our friends at the movies, we could find the total amount of money they need like this:

\begin{matrix} 3 (9.25) \\ \begin{matrix} 3 (9 & + & 0.25) \\ 3 (9) & + & 3 (0.25) \\ 27 & + & 0.75 \end{matrix} \\ 27.75 \end{matrix}

In algebra, we use the distributive property to remove parentheses as we simplify expressions.

For example, if we are asked to simplify the expression $3 (x + 4),$ the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the distributive property, as shown in Example 11.

Simplify: $3 (x + 4) .$

Solution

This table illustrates the step-by-step simplification of the algebraic expression 3(x+4) using the distributive property.
	$3 (x + 4)$
Distribute.	$3 \cdot x + 3 \cdot 4$
Multiply.	$3 x + 12$

Some students find it helpful to draw in arrows to remind them how to use the distributive property. Then the first step in Example 11 would look like this:

We have the expression 3 times (x plus 4) with two arrows coming from the 3. One arrow points to the x, and the other arrow points to the 4.

Simplify: $8 (\frac{3}{8} x + \frac{1}{4}) .$

Solution


Distribute.
Multiply.

Using the distributive property as shown in Example 13 will be very useful when we solve money applications in later chapters.

Simplify: $100 (0.3 + 0.25 q) .$

Solution


Distribute.
Multiply.

When we distribute a negative number, we need to be extra careful to get the signs correct!

Simplify: $−2 (4 y + 1) .$

Solution


Distribute.
Multiply.

Simplify: $−11 (4 - 3 a) .$

Solution

Distribute.
Multiply.
Simplify.

Notice that you could also write the result as $33 a - 44 .$ Do you know why?

Example 16 will show how to use the distributive property to find the opposite of an expression.

Simplify: $− (y + 5) .$

Solution

Step-by-step demonstration of multiplying (y+5) by -1, showing distribution and simplification.
	$(y + 5)$
Multiplying by −1 results in the opposite.	$−1 (y + 5)$
Distribute.	$−1 \cdot y + (−1) \cdot 5$
Simplify.	$− y + (−5)$
	$− y - 5$

There will be times when we’ll need to use the distributive property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

Simplify: $8 - 2 (x + 3) .$

Be sure to follow the order of operations. Multiplication comes before subtraction, so we will distribute the 2 first and then subtract.

Solution

Step-by-step simplification of the algebraic expression 8 - 2(x + 3).
	$8 - 2 (x + 3)$
Distribute.	$8 - 2 \cdot x - 2 \cdot 3$
Multiply.	$8 - 2 x - 6$
Combine like terms.	$−2 x + 2$

Simplify: $4 (x - 8) - (x + 3) .$

Solution

Demonstrates the step-by-step simplification of an algebraic expression using distribution and combining like terms.
	$4 (x - 8) - (x + 3)$
Distribute.	$4 x - 32 - x - 3$
Combine like terms.	$3 x - 35$

All the properties of real numbers we have used in this chapter are summarized in Table 22.

Commutative Property
of addition If $a, b$ are real numbers, then of multiplication If $a, b$ are real numbers, then	$a + b = b + a$ $a \cdot b = b \cdot a$
Associative Property
of addition If $a, b, c$ are real numbers, then of multiplication If $a, b, c$ are real numbers, then	$(a + b) + c = a + (b + c)$ $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
Distributive Property
If $a, b, c$ are real numbers, then	$a (b + c) = a b + a c$
Identity Property
of addition For any real number $a :$ 0 is the additive identity of multiplication For any real number $a :$ $1$ is the multiplicative identity	$\begin{matrix} a + 0 = a \\ 0 + a = a \end{matrix}$ $\begin{matrix} a \cdot 1 = a \\ 1 \cdot a = a \end{matrix}$
Inverse Property
of addition For any real number $a,$ $− a$ is the additive inverse of $a$ of multiplication For any real number $a, a \neq 0$ $\frac{1}{a}$ is the multiplicative inverse of $a .$	$a + (− a) = 0$ $a \cdot \frac{1}{a} = 1$
Properties of Zero
For any real number a, For any real number $a, a \neq 0$ For any real number $a, a \neq 0$	$\begin{matrix} a \cdot 0 = 0 \\ 0 \cdot a = 0 \end{matrix}$ $\frac{0}{a} = 0$ $\frac{a}{0}$ is undefined

Key Concepts

Commutative Property of
- Addition: If $a, b$ are real numbers, then $a + b = b + a .$
- Multiplication: If $a, b$ are real numbers, then $a \cdot b = b \cdot a .$ When adding or multiplying, changing the order gives the same result.
Associative Property of
- Addition: If $a, b, c$ are real numbers, then $(a + b) + c = a + (b + c) .$
- Multiplication: If $a, b, c$ are real numbers, then $(a \cdot b) \cdot c = a \cdot (b \cdot c) .$
  When adding or multiplying, changing the grouping gives the same result.
Distributive Property: If a,b,c are real numbers, then
- $a (b + c) = a b + a c$
- $(b + c) a = b a + c a$
- $a (b - c) = a b - a c$
- $(b - c) a = b a - c a$
Identity Property
- of Addition: For any real number $a : a + 0 = a 0 + a = a$
  0 is the additive identity
- of Multiplication: For any real number $a : a \cdot 1 = a 1 \cdot a = a$
  $1$ is the multiplicative identity
Inverse Property
- of Addition: For any real number $a, a + (− a) = 0 .$ A number and its opposite add to zero. $− a$ is the additive inverse of $a .$
- of Multiplication: For any real number $a, (a \neq 0) a \cdot \frac{1}{a} = 1 .$ A number and its reciprocal multiply to one. $\frac{1}{a}$ is the multiplicative inverse of $a .$
Properties of Zero
- For any real number $a,$
  $a \cdot 0 = 0 0 \cdot a = 0$ – The product of any real number and 0 is 0.
- $\frac{0}{a} = 0$ for $a \neq 0$ – Zero divided by any real number except zero is zero.
- $\frac{a}{0}$ is undefined – Division by zero is undefined.

Practice Makes Perfect

Use the Commutative and Associative Properties

In the following exercises, use the associative property to simplify.

3(4x)

Solution

12x

4(7m)

$(y + 12) + 28$

Solution

$y + 40$

$(n + 17) + 33$

In the following exercises, simplify.

$\frac{1}{2} + \frac{7}{8} + (- \frac{1}{2})$

Solution

$\frac{7}{8}$

$\frac{2}{5} + \frac{5}{12} + (- \frac{2}{5})$

$\frac{3}{20} \cdot \frac{49}{11} \cdot \frac{20}{3}$

Solution

$\frac{49}{11}$

$\frac{13}{18} \cdot \frac{25}{7} \cdot \frac{18}{13}$

$−24 \cdot 7 \cdot \frac{3}{8}$

Solution

$−63$

$−36 \cdot 11 \cdot \frac{4}{9}$

$(\frac{5}{6} + \frac{8}{15}) + \frac{7}{15}$

Solution

$1 \frac{5}{6}$

$(\frac{11}{12} + \frac{4}{9}) + \frac{5}{9}$

17(0.25)(4)

Solution

36(0.2)(5)

[2.48(12)](0.5)

Solution

14.88

[9.731(4)](0.75)

7(4a)

Solution

28a

9(8w)

$−15 (5 m)$

Solution

$−75 m$

$−23 (2 n)$

$12 (\frac{5}{6} p)$

Solution

10p

$20 (\frac{3}{5} q)$

$43 m + (−12 n) + (−16 m) + (−9 n)$

Solution

$27 m + (−21 n)$

$−22 p + 17 q + (−35 p) + (−27 q)$

$\frac{3}{8} g + \frac{1}{12} h + \frac{7}{8} g + \frac{5}{12} h$

Solution

$\frac{5}{4} g + \frac{1}{2} h$

$\frac{5}{6} a + \frac{3}{10} b + \frac{1}{6} a + \frac{9}{10} b$

$6.8 p + 9.14 q + (−4.37 p) + (−0.88 q)$

Solution

$2.43 p + 8.26 q$

$9.6 m + 7.22 n + (−2.19 m) + (−0.65 n)$

Use the Identity and Inverse Properties of Addition and Multiplication

In the following exercises, find the additive inverse of each number.

ⓐ $\frac{2}{5}$ ⓑ 4.3
ⓒ $−8$
ⓓ $- \frac{10}{3}$

Solution

ⓐ $- \frac{2}{5}$ ⓑ $−4.3$ ⓒ 8 ⓓ $\frac{10}{3}$

ⓐ $\frac{5}{9}$
ⓑ 2.1 ⓒ $−3$ ⓓ $- \frac{9}{5}$

ⓐ $- \frac{7}{6}$ ⓑ $−0.075$ ⓒ 23 ⓓ $\frac{1}{4}$

Solution

ⓐ $\frac{7}{6}$ ⓑ 0.075 ⓒ $−23$ ⓓ $- \frac{1}{4}$

ⓐ $- \frac{8}{3}$ ⓑ $−0.019$ ⓒ 52 ⓓ $\frac{5}{6}$

In the following exercises, find the multiplicative inverse of each number.

ⓐ 6 ⓑ $- \frac{3}{4}$ ⓒ 0.7

Solution

ⓐ $\frac{1}{6}$ ⓑ $- \frac{4}{3}$ ⓒ $\frac{10}{7}$

ⓐ 12 ⓑ $- \frac{9}{2}$ ⓒ 0.13

ⓐ $\frac{11}{12}$ ⓑ $−1.1$ ⓒ $−4$

Solution

ⓐ $\frac{12}{11}$ ⓑ $- \frac{10}{11}$ ⓒ $- \frac{1}{4}$

ⓐ $\frac{17}{20}$ ⓑ $−1.5$ ⓒ $−3$

Use the Properties of Zero

In the following exercises, simplify.

$\frac{0}{6}$

Solution

$\frac{3}{0}$

$0 \div \frac{11}{12}$

Solution

$\frac{6}{0}$

$\frac{0}{3}$

Solution

$0 \cdot \frac{8}{15}$

$(−3.14) (0)$

Solution

$\frac{\frac{1}{10}}{0}$

Mixed Practice

In the following exercises, simplify.

$19 a + 44 - 19 a$

Solution

$27 c + 16 - 27 c$

$10 (0.1 d)$

Solution

$100 (0.01 p)$

$\frac{0}{u - 4.99},$ where $u \neq 4.99$

Solution

$\frac{0}{v - 65.1},$ where $v \neq 65.1$

$0 \div (x - \frac{1}{2}),$ where $x \neq \frac{1}{2}$

Solution

$0 \div (y - \frac{1}{6}),$ where $y \neq \frac{1}{6}$

$\frac{32 - 5 a}{0},$ where
$32 - 5 a \neq 0$

Solution

undefined

$\frac{28 - 9 b}{0},$ where
$28 - 9 b \neq 0$

$(\frac{3}{4} + \frac{9}{10} m) \div 0$ where
$\frac{3}{4} + \frac{9}{10} m \neq 0$

Solution

undefined

$(\frac{5}{16} n - \frac{3}{7}) \div 0$ where
$\frac{5}{16} n - \frac{3}{7} \neq 0$

$15 \cdot \frac{3}{5} (4 d + 10)$

Solution

$36 d + 90$

$18 \cdot \frac{5}{6} (15 h + 24)$

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

$8 (4 y + 9)$

Solution

$32 y + 72$

$9 (3 w + 7)$

$6 (c - 13)$

Solution

$6 c - 78$

$7 (y - 13)$

$\frac{1}{4} (3 q + 12)$

Solution

$\frac{3}{4} q + 3$

$\frac{1}{5} (4 m + 20)$

$9 (\frac{5}{9} y - \frac{1}{3})$

Solution

$5 y - 3$

$10 (\frac{3}{10} x - \frac{2}{5})$

$12 (\frac{1}{4} + \frac{2}{3} r)$

Solution

$3 + 8 r$

$12 (\frac{1}{6} + \frac{3}{4} s)$

$r (s - 18)$

Solution

$r s - 18 r$

$u (v - 10)$

$(y + 4) p$

Solution

$y p + 4 p$

$(a + 7) x$

$−7 (4 p + 1)$

Solution

$−28 p - 7$

$−9 (9 a + 4)$

$−3 (x - 6)$

Solution

$−3 x + 18$

$−4 (q - 7)$

$− (3 x - 7)$

Solution

$−3 x + 7$

$− (5 p - 4)$

$16 - 3 (y + 8)$

Solution

$−3 y - 8$

$18 - 4 (x + 2)$

$4 - 11 (3 c - 2)$

Solution

$−33 c + 26$

$9 - 6 (7 n - 5)$

$22 - (a + 3)$

Solution

$− a + 19$

$8 - (r - 7)$

$(5 m - 3) - (m + 7)$

Solution

$4 m - 10$

$(4 y - 1) - (y - 2)$

$5 (2 n + 9) + 12 (n - 3)$

Solution

$22 n + 9$

$9 (5 u + 8) + 2 (u - 6)$

$9 (8 x - 3) - (−2)$

Solution

$72 x - 25$

$4 (6 x - 1) - (−8)$

$14 (c - 1) - 8 (c - 6)$

Solution

$6 c + 34$

$11 (n - 7) - 5 (n - 1)$

$6 (7 y + 8) - (30 y - 15)$

Solution

$12 y + 63$

$7 (3 n + 9) - (4 n - 13)$

Everyday Math

Insurance copayment Carrie had to have 5 fillings done. Each filling cost $80. Her dental insurance required her to pay 20% of the cost as a copay. Calculate Carrie’s copay:

ⓐ First, by multiplying 0.20 by 80 to find her copay for each filling and then multiplying your answer by 5 to find her total copay for 5 fillings.
ⓑ Next, by multiplying [5(0.20)](80)
ⓒ Which of the properties of real numbers says that your answers to parts (a), where you multiplied 5[(0.20)(80)] and (b), where you multiplied [5(0.20)](80), should be equal?

Solution

Cooking time Helen bought a 24-pound turkey for her family’s Thanksgiving dinner and wants to know what time to put the turkey in to the oven. She wants to allow 20 minutes per pound cooking time. Calculate the length of time needed to roast the turkey:

ⓐ First, by multiplying $24 \cdot 20$ to find the total number of minutes and then multiplying the answer by $\frac{1}{60}$ to convert minutes into hours.
ⓑ Next, by multiplying $24 (20 \cdot \frac{1}{60}) .$
ⓒ Which of the properties of real numbers says that your answers to parts (a), where you multiplied $(24 \cdot 20) \frac{1}{60},$ and (b), where you multiplied $24 (20 \cdot \frac{1}{60}),$ should be equal?

Buying by the case Trader Joe’s grocery stores sold a bottle of wine they called “Two Buck Chuck” for $1.99. They sold a case of 12 bottles for $23.88. To find the cost of 12 bottles at $1.99, notice that 1.99 is $2 - 0.01 .$

ⓐ Multiply 12(1.99) by using the distributive property to multiply $12 (2 - 0.01) .$
ⓑ Was it a bargain to buy “Two Buck Chuck” by the case?

Solution

ⓐ $23.88 ⓑ no, the price is the same

Multi-pack purchase Adele’s shampoo sells for $3.99 per bottle at the grocery store. At the warehouse store, the same shampoo is sold as a 3 pack for $10.49. To find the cost of 3 bottles at $3.99, notice that 3.99 is $4 - 0.01 .$

ⓐ Multiply 3(3.99) by using the distributive property to multiply $3 (4 - 0.01) .$
ⓑ How much would Adele save by buying 3 bottles at the warehouse store instead of at the grocery store?

Writing Exercises

In your own words, state the commutative property of addition.

Solution

Answers may vary

What is the difference between the additive inverse and the multiplicative inverse of a number?

Simplify $8 (x - \frac{1}{4})$ using the distributive property and explain each step.

Solution

Answers may vary

Explain how you can multiply 4($5.97) without paper or calculator by thinking of $5.97 as $6 - 0.03$ and then using the distributive property.

Self Check

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

This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “use the commutative and associative properties,” “use the identity and inverse properties of addition and multiplication,” “use the properties of zero,” and “simplify expressions using the distributive property.” The rest of the cells are blank.

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

additive identity

The additive identity is the number 0; adding 0 to any number does not change its value.

additive inverse

The opposite of a number is its additive inverse. A number and its additive inverse add to 0.

multiplicative identity

The multiplicative identity is the number 1; multiplying 1 by any number does not change the value of the number.

multiplicative inverse

The reciprocal of a number is its multiplicative inverse. A number and its multiplicative inverse multiply to one.