Question

which expression demonstrates the use of the commutative property of addition in the first step of simplifying the expression (-1 + i) + (21 + 5i)?
○ (-1 + i) + (21 + 5i) + 0
○ -1 + (i + 21) + 5i
○ (-1 + 21) + (i + 5i)
○ -(1 - i) + (21 + 5i)

Explanation:

Step1: Recall Commutative Property of Addition

The commutative property of addition states that for any two numbers \(a\) and \(b\), \(a + b=b + a\). In the context of adding complex numbers (or expressions with terms), we can rearrange the order of addition of the terms. The original expression is \((-1 + i)+(21 + 5i)\). We can think of the terms as \(-1\), \(i\), \(21\), and \(5i\). We want to rearrange the order of addition to group the real parts and the imaginary parts together.

Step2: Analyze Each Option

• Option 1: \((-1 + i)+(21 + 5i)+0\) adds a zero, which is related to the identity property, not commutative.
• Option 2: \(-1+(i + 21)+5i\) uses the associative property (grouping \(i\) and \(21\)) but not commutative in the first step (the order of \(i\) and \(21\) is changed, but the main commutative step for grouping real and imaginary isn't done here as well as option 3).
• Option 3: \((-1 + 21)+(i + 5i)\) rearranges the terms so that the real parts \(-1\) and \(21\) are added together and the imaginary parts \(i\) and \(5i\) are added together. This is done by using the commutative property to swap the order of \(i\) and \(21\) (from \((-1 + i)+(21 + 5i)\) to \((-1+21)+(i + 5i)\) by commuting \(i\) and \(21\) and then associating).
• Option 4: \(-(1 - i)+(21 + 5i)\) is a sign change, not related to commutative property of addition.

Answer:

\((-1 + 21)+(i + 5i)\) (the third option)