Which expression demonstrates the use of the commutative property of addition in the first step of simplifying the expression $(-1+i)+(21+5 i)$?
A. $(-1+i)+(21+5 i)+0$
B. $-1+(i+21)+5 i$
C. $(-1+21)+(i+5 i)$
D. $-(1-i)+(21+5 i)$
Answer (2)
The problem requires identifying the expression that uses the commutative property of addition in the first step of simplifying ( − 1 + i ) + ( 21 + 5 i ) .
The commutative property states that a + b = b + a .
The expression ( − 1 + 21 ) + ( i + 5 i ) correctly applies the commutative property by grouping real and imaginary parts.
Therefore, the answer is ( − 1 + 21 ) + ( i + 5 i ) .
Explanation
Understanding the Commutative Property The problem asks us to identify which expression demonstrates the use of the commutative property of addition as the first step in simplifying ( − 1 + i ) + ( 21 + 5 i ) . The commutative property states that a + b = b + a for any numbers a and b . We need to see which of the options correctly applies this property in the initial step.
Analyzing the Options Let's examine each option:
( − 1 + i ) + ( 21 + 5 i ) + 0 : This adds 0, which is the identity property, not the commutative property.
− 1 + ( i + 21 ) + 5 i : This uses the associative property, where the grouping of terms is changed, not the order of terms.
( − 1 + 21 ) + ( i + 5 i ) : This expression rearranges and regroups the terms. To get here, we can use the commutative property to rewrite the original expression as ( − 1 + i ) + ( 21 + 5 i ) = ( − 1 + 21 ) + ( i + 5 i ) .
− ( 1 − i ) + ( 21 + 5 i ) : This changes the sign of the first term, which is not the commutative property.
Identifying the Correct Expression The correct application of the commutative property involves changing the order of the terms being added. In the expression ( − 1 + i ) + ( 21 + 5 i ) , we want to rearrange the terms such that real parts and imaginary parts are grouped together. This is achieved by ( − 1 + 21 ) + ( i + 5 i ) .
Final Answer Therefore, the expression that demonstrates the use of the commutative property of addition in the first step of simplifying the given expression is ( − 1 + 21 ) + ( i + 5 i ) .
Examples
The commutative property is useful in everyday situations, such as when calculating the total cost of items in a store. If you buy a shirt for $25 and pants for $40, the total cost is the same whether you add $25 + $40 or $40 + $25. This property simplifies mental calculations and ensures accuracy regardless of the order in which you add the numbers.
The expression that demonstrates the commutative property of addition in simplifying ( − 1 + i ) + ( 21 + 5 i ) is ( − 1 + 21 ) + ( i + 5 i ) . This option correctly rearranges the terms while maintaining their sum. Therefore, the selected answer is Option C: ( − 1 + 21 ) + ( i + 5 i ) .
;
