Problem 32 - Elementary and Intermediate Algebra

Question

Use the associative law of addition to write an equivalent expression. $$ (m+n p)+r $$

Step-by-Step Solution

Answer

m + (n p + r)

Step 1 - Identify the Associative Law of Addition

The associative law of addition states that the way in which numbers are grouped in addition does not change their sum. Mathematically, it is represented as \(a + (b + c) = (a + b) + c\).

Step 2 - Apply the Associative Law to the Given Expression

Given the expression \((m + n p) + r\), recognize that we can regroup the numbers without changing the sum. By applying the associative law of addition, rewrite the expression as \(m + (n p + r)\).

Key Concepts

Addition Properties

In mathematics, addition properties help simplify complex problems and make calculations easier. These properties include:

• Commutative Property: This property states that the order of adding numbers does not affect the sum. For instance, \( a + b = b + a \).
• Associative Property: This property states that the grouping of numbers does not change their sum. For example, \( a + (b + c) = (a + b) + c \).
• Identity Property: This property states that adding zero to any number does not change its value. For example, \( a + 0 = a \).

Let’s focus on the associative property and see how it simplifies algebraic expressions.”

Algebraic Expressions

Algebraic expressions consist of variables (letters), numbers, and operations like addition or multiplication. They allow us to represent real-world problems and abstract ideas mathematically. In an expression:

• Variables such as \( m \), \( n \), and \( r \) represent unknown values.
• Constants are fixed numbers.
• Coefficients are numbers that multiply the variables.

Understanding how to manipulate and simplify these expressions using properties like the associative law is important for solving algebraic problems. For instance, the expression \((m + n p) + r\) can be regrouped using the associative property to \(m + (n p + r)\). This makes it easier to handle in larger calculations or equations.

Grouping in Addition

Grouping in addition refers to the practice of rearranging and regrouping numbers or variables to simplify expressions. The associative property is essential for this because it tells us that the sum remains unchanged regardless of how we group the numbers. For example:

• Given \( (m + n p) + r \), applying the associative property lets us regroup to \( m + (n p + r) \).
• This rearrangement can make further simplification or computation more straightforward.
• It does not change the sum and helps us to see relationships between numbers or terms more clearly.

Grouping in addition streamlines problem-solving and ensures clarity in more complex algebraic expressions.