Which property was used to simplify the expression?
[tex]4(b+2)=4 b+8[/tex]
A. distributive property
B. commutative property
C. associative property
D. inverse property
Answer (2)
The distributive property is used to simplify the expression 4 ( b + 2 ) = 4 b + 8 . The distributive property states that a ( b + c ) = ab + a c . In this case, 4 ( b + 2 ) = 4 ⋅ b + 4 ⋅ 2 = 4 b + 8 . Therefore, the answer is distributive property.
Explanation
Understanding the Problem We are asked to identify the property used to simplify the expression 4 ( b + 2 ) = 4 b + 8 . Let's review the properties listed as options.
Distributive Property The distributive property states that a ( b + c ) = ab + a c . This property involves multiplying a term by a sum (or difference) of terms inside parentheses.
Commutative Property The commutative property states that the order of operations does not matter. For addition, a + b = b + a , and for multiplication, ab = ba .
Associative Property The associative property states that the grouping of terms does not matter. For addition, ( a + b ) + c = a + ( b + c ) , and for multiplication, ( ab ) c = a ( b c ) .
Inverse Property The inverse property states that for every number a , there exists an additive inverse − a such that a + ( − a ) = 0 , and for every nonzero number a , there exists a multiplicative inverse a 1 such that a ⋅ a 1 = 1 .
Identifying the Property Comparing the given expression 4 ( b + 2 ) = 4 b + 8 with the properties, we see that it matches the distributive property, where a = 4 , b = b , and c = 2 . Specifically, 4 ( b + 2 ) = 4 ⋅ b + 4 ⋅ 2 = 4 b + 8 .
Examples
The distributive property is commonly used in everyday life when calculating costs. For example, if you want to buy 4 items that each cost b + 2 dollars, the total cost is 4 ( b + 2 ) . Using the distributive property, you can calculate this as 4 b + 8 , which means you are paying 4 b dollars for the items plus an additional 8 dollars.
The expression 4 ( b + 2 ) = 4 b + 8 was simplified using the distributive property, which allows a number to multiply each term inside parentheses. By applying this property, we distribute 4 to both b and 2 , resulting in 4 b + 8 . Therefore, the correct answer is A. distributive property.
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