Which expression demonstrates the use of the commutative property of addition in the first step of simplifying the expression $(-1+i)+(21+5i)$?
A. $(-1+i)+(21+5i)+0$
B. $-1+(i+21)+5i$
C. $(-1+21)+(i+5i)$
D. $-(1-i)+(21+5i)$
Answer (2)
The expression that demonstrates the use of the commutative property of addition is ( − 1 + 21 ) + ( i + 5 i ) . This option shows how to rearrange the terms correctly by grouping real and imaginary parts. Therefore, the correct choice is Option C .
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The problem requires identifying the expression that uses the commutative property of addition in the first step of simplifying ( − 1 + A + ( 21 + E A ) .
The commutative property states that a + b = b + a .
Substituting A = i and E = 5 into the original expression gives ( − 1 + i + ( 21 + 5 i )) .
Applying the commutative property, the expression becomes ( − 1 + 21 ) + ( i + 5 i ) , which is the correct answer: ( − 1 + 21 ) + ( i + 5 i ) .
Explanation
Understanding the Problem We are asked to identify the expression that demonstrates the use of the commutative property of addition in the first step of simplifying the expression ( − 1 + A + ( 21 + E A ) . The commutative property of addition states that a + b = b + a for any numbers a and b .
Applying Commutative Property Let's analyze the given expression: ( − 1 + A + ( 21 + E A ) . We want to rearrange the terms using the commutative property in the first step. The commutative property allows us to change the order of addition.
Analyzing the Options Now, let's examine each option:
( − 1 + i ) + ( 21 + 5 i ) + 0 : This option substitutes A with i and E with 5 , and adds 0. It doesn't directly demonstrate the commutative property in the first step.
− 1 + ( i + 21 ) + 5 i : This option substitutes A with i and E with 5 , and uses the associative property. It doesn't demonstrate the commutative property in the first step.
( − 1 + 21 ) + ( i + 5 i ) : This option substitutes A with i and E with 5 , and uses both the commutative and associative properties. The commutative property is used to rearrange the terms to group the real numbers and the imaginary numbers together. This is a possible answer.
− ( 1 − i ) + ( 21 + 5 i ) : This option substitutes A with i and E with 5 , and changes the sign of the first term. It doesn't demonstrate the commutative property.
Identifying the Correct Expression The original expression is ( − 1 + A + ( 21 + E A ) . If we substitute A = i and E = 5 , we get ( − 1 + i + ( 21 + 5 i )) . Using the commutative property, we can rearrange the terms as ( − 1 + 21 + i + 5 i ) , which can be grouped as ( − 1 + 21 ) + ( i + 5 i ) . This matches option 3.
Conclusion Therefore, the expression that demonstrates the use of the commutative property of addition in the first step is ( − 1 + 21 ) + ( i + 5 i ) .
Examples
The commutative property is useful in everyday situations. For example, when calculating the total cost of items in a shopping cart, the order in which you add the prices doesn't change the final amount. If you buy a book for $15 and a pen for $5, the total cost is $15 + $5 = $20, which is the same as $5 + $15 = $20. This principle applies to various scenarios, such as managing budgets, calculating distances, or combining ingredients in a recipe.
