Lesson Objectives
• Learn about the identity properties
• Learn about the inverse properties
• Learn about the distributive property

## Commutative Property

The commutative property allows us to reorder our addition and multiplication problems. The commutative property of addition tells us that the order in which we add two or more numbers will never change the result. Similarly, the commutative property of multiplication tells us that the order in which we multiply two or more numbers will never change the result. $$a + b=b + a$$ $$a \cdot b=b \cdot a$$ Example 1: Rewrite each using the commutative property $$4 + (-3)$$ To complete the problem, we just rearrange the order of the addends or numbers being added together. $$4 + (-3)=(-3) + 4$$ $$4 + (-3)=1$$ $$(-3) + 4=1$$ Changing the order did not change the sum. Either way, the result is 1.
Example 2: Rewrite each using the commutative property $$-5 \cdot 14$$ To complete the problem, we just rearrange the order of the factors or numbers being multiplied together. $$-5 \cdot 14=14 \cdot -5$$ $$-5 \cdot 14=-70$$ $$14 \cdot -5=-70$$ Changing the order did not change the product. Either way, the result is -70.

## Associative Property

The associative property deals with how we group together three or more numbers for addition or multiplication. The associative property of addition tells us that we can group the addition of three or more numbers in any manner and never change the result (sum). Similarly, the associative property of multiplication tells us that we can group the multiplication of three or more numbers in any manner and never change the result (product). $$a + (b + c)=(a + b) + c$$ $$a \cdot (b \cdot c)=(a \cdot b) \cdot c$$ Example 3: Rewrite each using the associative property $$(3 + 5) + 1$$ To complete the problem, we just change the grouping or the operation that is enclosed by parentheses. $$(3 + 5) + 1=3 + (5 + 1)$$ $$(3 + 5) + 1=9$$ $$3 + (5 + 1)=9$$ Changing the grouping did not change the sum. Either way, the result is 9.
Example 4: Rewrite each using the associative property $$9 \cdot (3 \cdot 2)$$ To complete the problem, we just change the grouping or the operation that is enclosed by parentheses. $$9 \cdot (3 \cdot 2)=(9 \cdot 3) \cdot 2$$ $$9 \cdot (3 \cdot 2)=9 \cdot 6=54$$ $$(9 \cdot 3) \cdot 2=27 \cdot 54$$ Changing the grouping did not change the product. Either way, the result is 9.

## Identity Properties

Zero is known as the additive identity. Adding zero to any number will leave the number unchanged. $$a + 0=a$$ $$1075 + 0=1075$$ $$-51 + 0=-51$$ One is known as the multiplicative identity. Multiplying a nonzero number by 1 will leave the number unchanged. Remember, multiplying by zero will always result in an answer of zero. This is why we say "nonzero" number. $$a \cdot 1=a, a ≠ 0$$ $$-5 \cdot 1=-5$$ $$\frac{1}{3}\cdot 1=\frac{1}{3}$$ Example 5: Perform each indicated operation $$110 + 0$$ Adding zero to a number will not change the number. $$110 + 0=110$$ Example 6: Perform each indicated operation $$-519 \cdot 1$$ Multiplying a nonzero number by 1 will not change the number. $$-519 \cdot 1=-519$$

## Inverse Properties

When a number and its opposite (additive inverse) are added together, the result is always zero. Remember the opposite or additive inverse of a number is found by just changing the sign. $$a + (-a)=0$$ $$-22 + 22=0$$ $$1055 + (-1055)=0$$ Additionally, when a nonzero number and its reciprocal are multiplied together, the result is always 1. $$a \cdot \frac{1}{a}=1, a ≠ 0$$ $$-4 \cdot \frac{1}{-4}$$ $$\frac{3}{2}\cdot \frac{2}{3}=1$$ Example 7: Perform each indicated operation $$414 + (-414)$$ Adding a number and its opposite will always result in zero. $$414 + (-414)=0$$ Example 8: Perform each indicated operation $$\frac{1}{23}\cdot \frac{23}{1}$$ Multiplying a nonzero number by its reciprocal will always result in 1. $$\frac{1}{23}\cdot \frac{23}{1}=1$$

## Distributive Property

The distributive property is extremely important. It allows us to change a sum into a product or a product into a sum. $$a(b + c)=ab + ac$$ $$ab + ac=a(b + c)$$ Basically, the distributive property tells us that we can distribute multiplication over addition or subtraction. Additionally, we can reverse the process and pull the factor back out, which is known as factoring. $$3(x + 6)=3 \cdot x + 3 \cdot 6=3x + 18$$ $$5x + 25=5 \cdot x + 5 \cdot 5=5(x + 5)$$ Example 9: Use the distributive property to simplify $$3(x + y + 2)$$ To solve this problem, we can distribute the 3 to each term inside of the parentheses. $$3(x + y + 2)=3x + 3y + 6$$ Example 10: Use the distribute property to factor $$3x + 21$$ To solve this problem, we can think about what is common to both terms. $$\require{color}\colorbox{yellow}{3}\cdot x + \colorbox{yellow}{3}\cdot 7$$ Since 3 is common to both terms, this can be factored out. Wrap the expression with a set of parentheses. $$\require{color}(\colorbox{yellow}{3}\cdot x + \colorbox{yellow}{3}\cdot 7)$$ Pull out the 3 that is common to each term and place this in front of the parentheses. $$3(x + 7)$$

#### Skills Check:

Example #1

Determine which statement is true.

A
$$5 \cdot (-5)=1$$
B
$$4 + (-4)=\frac{1}{4}$$
C
$$\frac{6}{5}\cdot -\frac{5}{6}=1$$
D
$$13 \cdot \frac{1}{13}=1$$
E
$$91 + (-91)=\frac{1}{91}$$

Example #2

Determine which statement is true.

A
$$2(x + 3)=2(2x + 6)$$
B
$$-12 + 12=0$$
C
$$18x - 20=18(x - 2)$$
D
$$9 \cdot -\frac{1}{9}=0$$
E
$$114 + 0=0$$

Example #3

Determine which statement is true.

A
$$-\frac{1}{3}\cdot -3=1$$
B
$$-103 + 103=1$$
C
$$-232 \cdot 0=1$$
D
$$5 + 9=9 \cdot 5$$
E
$$3 + (2 + 7)=(3 \cdot 2) \cdot 7$$

Example #4

Determine which statement is true.

A
$$15(x - 3)=15x + 45$$
B
$$\frac{1}{9}\cdot 1=0$$
C
$$332 + 0=-332$$
D
$$(8 + 3) + 2=8 \cdot (3 \cdot 2)$$
E
$$1 + (-9)=-9 + 1$$