Which equation shows an example of the associative property of addition?
A. [tex]$4 i \cdot(-4 i+i)=(4 i-4 i)+(4 i \cdot i)$[/tex]
B. [tex]$(-4 i+i)+0=(-4 i+i)$[/tex]
C. [tex]$(-4+i)+4 i=-4+(i+4 i)$[/tex]
D. [tex]$(-4+i)+4 i=4 i+(-4 i+i)$[/tex]
Answer (2)
The associative property of addition states that ( a + b ) + c = a + ( b + c ) .
Examine each equation to see if it fits the associative property.
The equation ( − 4 + i ) + 4 i = − 4 + ( i + 4 i ) demonstrates the associative property.
Therefore, the correct equation is ( − 4 + i ) + 4 i = − 4 + ( i + 4 i ) .
Explanation
Understanding the Associative Property The question asks us to identify which equation demonstrates the associative property of addition. The associative property states that how numbers are grouped when adding does not change the sum, i.e., ( a + b ) + c = a + ( b + c ) . We need to examine each equation to see if it fits this form.
Analyzing the Equations Let's analyze the given equations:
4 i ⋅ ( − 4 i + i ) = ( 4 i − 4 i ) + ( 4 i ⋅ i ) . This equation involves multiplication, so it's not the associative property of addition.
( − 4 i + i ) + 0 = ( − 4 i + i ) . This equation shows the identity property of addition (adding zero doesn't change the value), not the associative property.
( − 4 + i ) + 4 i = − 4 + ( i + 4 i ) . This equation fits the form ( a + b ) + c = a + ( b + c ) , where a = − 4 , b = i , and c = 4 i . This demonstrates the associative property of addition.
( − 4 + i ) + 4 i = 4 i + ( − 4 + i ) . This equation demonstrates the commutative property of addition ( a + b = b + a ), not the associative property.
Identifying the Correct Equation The equation that demonstrates the associative property of addition is ( − 4 + i ) + 4 i = − 4 + ( i + 4 i ) .
Examples
The associative property of addition is useful in many real-life situations. For example, when calculating the total cost of items in a shopping cart, you can group the items in any order and still arrive at the same total. If you buy a book for $10, a pen for $2, and a notebook for 5 , yo u c an c a l c u l a t e t h e t o t a l a s (10 + 2) + 5 = 12 + 5 = 17$ or as 10 + ( 2 + 5 ) = 10 + 7 = 17 . The associative property ensures that the order in which you add the costs doesn't affect the final amount.
The equation that demonstrates the associative property of addition is C: ( − 4 + i ) + 4 i = − 4 + ( i + 4 i ) . This equation shows that grouping of the terms does not affect the sum. Other equations either do not involve addition or illustrate different properties.
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