Certainly! Let's walk through the given expression using the associative property to simplify it using mental math. We'll evaluate step by step and group different terms.
Firstly, let's state the expression:
[tex]\[ -25 + 12 + 13 + (-7) \][/tex]
### Option 1: Group [tex]\( 12 + 13 \)[/tex] together first.
[tex]\[ -25 + (12 + 13) + (-7) \][/tex]
- Calculate inside the parentheses first:
- [tex]\( 12 + 13 = 25 \)[/tex]
- Substitute back into the expression:
- [tex]\( -25 + 25 + (-7) \)[/tex]
- Combine [tex]\( -25 \)[/tex] and [tex]\( 25 \)[/tex]:
- [tex]\( 0 + (-7) = -7 \)[/tex]
### Option 2: Group [tex]\( (12 + 13) - (-7) \)[/tex] together first.
[tex]\[ -25 + (12 + 13) - (-7) \][/tex]
- Calculate inside the parentheses first:
- [tex]\( 12 + 13 = 25 \)[/tex]
- Substitute back into the equation:
- [tex]\( -25 + 25 + (-7) \)[/tex]
- Combine [tex]\( -25 \)[/tex] and [tex]\( 25 \)[/tex]:
- [tex]\( 0 + (-7) = -7 \)[/tex]
### Option 3: Group [tex]\( 13 + (-7) \)[/tex] together first.
[tex]\[ -25 + 12 + (13 + (-7)) \][/tex]
- Calculate inside the parentheses first:
- [tex]\( 13 + (-7) = 6 \)[/tex]
- Substitute back into the expression:
- [tex]\( -25 + 12 + 6 \)[/tex]
- Combine [tex]\( 12 \)[/tex] and [tex]\( 6 \)[/tex]:
- [tex]\( 18 \)[/tex]
- Finally, add [tex]\( -25 \)[/tex] and [tex]\( 18 \)[/tex]:
- [tex]\( 18 + (-25) = -7 \)[/tex]
### Option 4: Group [tex]\( (-25 + 12) + 13 + (-7) \)[/tex] together first.
[tex]\[ (-25 + 12) + 13 + (-7) \][/tex]
- Calculate inside the parentheses first:
- [tex]\( -25 + 12 = -13 \)[/tex]
- Substitute back into the equation:
- [tex]\( -13 + 13 + (-7) \)[/tex]
- Combine [tex]\( -13 \)[/tex] and [tex]\( 13 \)[/tex]:
- [tex]\( 0 + (-7) = -7 \)[/tex]
In all cases, simplifying the expression yields the same result:
[tex]\[ -7 \][/tex]
So, using the associative property, regardless of how the terms are grouped, the simplified solution of the expression is:
[tex]\[ -7 \][/tex]
