Certainly! Let's explore the given expression [tex]\((n + m) + 99\)[/tex] using the commutative and associative laws of addition.
### Commutative Law of Addition
The commutative law states that the order in which we add two numbers does not change the sum:
[tex]\[ a + b = b + a \][/tex]
Applying this to our expression:
[tex]\[ (n + m) + 99 \][/tex]
Let's switch the order inside the parentheses:
[tex]\[ (m + n) + 99 \][/tex]
This gives us our first equivalent expression:
[tex]\[
\text{Expression 1: } (m + n) + 99
\][/tex]
### Associative Law of Addition
The associative law states that how we group the numbers when we add does not change the sum:
[tex]\[ (a + b) + c = a + (b + c) \][/tex]
Using this law, we can regroup the numbers in our expression:
[tex]\[ (n + m) + 99 \][/tex]
Re-group the addition:
[tex]\[ n + (m + 99) \][/tex]
This gives us our second equivalent expression:
[tex]\[
\text{Expression 2: } n + (m + 99)
\][/tex]
To find another equivalent expression we can again use the commutative law, but this time on the term inside the parentheses of our second expression:
#### Applying Commutative Law inside Parentheses
Switch the order inside the parentheses of the second expression:
[tex]\[ n + (m + 99) \][/tex]
This can be expressed as:
[tex]\[ n + (99 + m) \][/tex]
By the commutative law we get:
[tex]\[ n + (99 + m) = n + (m + 99) \][/tex]
As this still represents our second expression, let's return to our original expression and apply the associative law again but in reverse:
[tex]\[ (n + m) + 99 \][/tex]
Switch the grouping again to:
[tex]\[ (m + 99) + n \][/tex]
By commutative law inside the parentheses:
[tex]\[ (99 + m) + n \][/tex]
This provides our third equivalent expression:
[tex]\[
\text{Expression 3: } (99 + m) + n
\][/tex]
### Summary
The three equivalent expressions using the commutative and associative laws of addition are:
1. [tex]\((m + n) + 99\)[/tex]
2. [tex]\(n + (m + 99)\)[/tex]
3. [tex]\((99 + m) + n\)[/tex]
