Problem 40
Question
Use the commutative and/or associative laws to write two equivalent expressions. Then simplify. Answers may vary. $$ (11+v)+4 $$
Step-by-Step Solution
Verified Answer
(11+v)+4 can be rewritten as 11+(v+4) or (v+11)+4. It is already in simplest form.
1Step 1: Identify the Expression
The given expression is (11+v)+4.
2Step 1: Apply the Associative Law
The Associative Law states that you can regroup terms with parentheses without changing the result. For this expression, we can regroup the terms differently: (11+v)+4 = 11+(v+4).
3Step 2: Apply the Commutative Law
The Commutative Law states that you can rearrange the terms without changing the result. In this case, we can also change the order of the terms: (11+v)+4 = (v+11)+4.
4Step 3: Simplify the Expression
Although the steps above show how to use the associative and commutative laws, the given expression (11+v)+4 is already in its simplest form. There are no like terms to combine or further simplification needed.
Key Concepts
Associative LawCommutative LawSimplification of Expressions
Associative Law
The Associative Law is a fundamental principle in algebra that dictates how you can group numbers when adding or multiplying.
A common form of the Associative Law for addition is: \((a + b) + c = a + (b + c)\).
In simpler terms, it means that the way in which numbers are grouped does not affect their sum.
Using the given exercise, we can apply the Associative Law to the expression \((11 + v) + 4\).
By regrouping the terms, the expression can be changed to \(11 + (v + 4)\).
Notice that despite regrouping the numbers or variables, the sum remains the same.
This demonstrates the Associative Law in action.
A common form of the Associative Law for addition is: \((a + b) + c = a + (b + c)\).
In simpler terms, it means that the way in which numbers are grouped does not affect their sum.
Using the given exercise, we can apply the Associative Law to the expression \((11 + v) + 4\).
By regrouping the terms, the expression can be changed to \(11 + (v + 4)\).
Notice that despite regrouping the numbers or variables, the sum remains the same.
This demonstrates the Associative Law in action.
Commutative Law
The Commutative Law allows you to rearrange the order of numbers or variables when adding or multiplying, without changing the result.
The Commutative Law for addition can be written as: \(a + b = b + a\).
This law highlights that the order in which you add numbers does not affect their sum.
For instance, in the exercise, the expression \((11 + v) + 4\) can be reorganized using the Commutative Law.
We rearrange the terms inside the parentheses: \((v + 11) + 4\).
Even though we changed the order of the terms, the sum will still be the same.
This flexibility makes it easier to simplify and solve expressions.
The Commutative Law for addition can be written as: \(a + b = b + a\).
This law highlights that the order in which you add numbers does not affect their sum.
For instance, in the exercise, the expression \((11 + v) + 4\) can be reorganized using the Commutative Law.
We rearrange the terms inside the parentheses: \((v + 11) + 4\).
Even though we changed the order of the terms, the sum will still be the same.
This flexibility makes it easier to simplify and solve expressions.
Simplification of Expressions
Simplification of expressions involves reducing them to their simplest form.
This could mean combining like terms, reducing fractions, or simply rearranging terms using the Associative and Commutative Laws.
In our given exercise, the expression \((11 + v) + 4\) is already simplified.
There are no like terms to combine, and it is expressed as simply as possible.
However, by using the Associative and Commutative Laws, we created equivalent forms, such as \(11 + (v + 4)\) and \((v + 11) + 4\).
These forms demonstrate that understanding these algebraic laws not only helps in balancing and solving equations but also in recognizing equivalent expressions.
This could mean combining like terms, reducing fractions, or simply rearranging terms using the Associative and Commutative Laws.
In our given exercise, the expression \((11 + v) + 4\) is already simplified.
There are no like terms to combine, and it is expressed as simply as possible.
However, by using the Associative and Commutative Laws, we created equivalent forms, such as \(11 + (v + 4)\) and \((v + 11) + 4\).
These forms demonstrate that understanding these algebraic laws not only helps in balancing and solving equations but also in recognizing equivalent expressions.