To determine which expression demonstrates the use of the commutative property of addition in the first step of simplifying [tex]\((-1+i)+(21+5i)\)[/tex], let's analyze each option given.
Recall that the commutative property of addition states that [tex]\(a + b = b + a\)[/tex]. We are looking for an expression where the order of the numbers being added is switched.
1. [tex]\((-1+i)+(21+5i)+0\)[/tex]
- Here, an extra 0 is added, which does not change the expression. The order of addition is not changed in any part of the expression.
2. [tex]\(-1+(i+21)+5i\)[/tex]
- This expression changes the grouping of terms but does not change the order. The terms [tex]\(-1\)[/tex], [tex]\(i\)[/tex], 21, and [tex]\(5i\)[/tex] are not reordered.
3. [tex]\((-1+21)+(i+5i)\)[/tex]
- In this expression, we see that the order of the terms is changed:
- [tex]\((-1)\)[/tex] and 21 are added together.
- [tex]\(i\)[/tex] and [tex]\(5i\)[/tex] are added together.
- This is the application of the commutative property; the components of the original addition have been reordered:
- [tex]\((-1)\)[/tex] and 21 are now paired.
- [tex]\(i\)[/tex] and [tex]\(5i\)[/tex] are now paired.
4. [tex]\(-(1-i)+(21+5i)\)[/tex]
- This expression involves a negation operation [tex]\(-(1-i)\)[/tex], which changes the signs of the terms but does not use the commutative property to switch the order of addition.
Given these analyses, the expression that correctly demonstrates the use of the commutative property of addition is:
[tex]\[(-1+21)+(i+5i)\][/tex]
Thus, the correct answer is:
[tex]\[ \text{Option 3:} (-1+21)+(i+5i) \][/tex]
