Problem 27 - Prealgebra

Question

Apply the associative property to expression, and then simplify the result. \((7 x+8)+20\)

Step-by-Step Solution

Answer

The simplified expression is \(7x + 28\).

Step 1: Identify the Terms

Look at the given expression \((7x + 8) + 20\).Here, we have a group \((7x + 8)\) and another term, 20. Our goal is to apply the associative property to simplify this expression.

Step 2: Apply the Associative Property

The associative property of addition allows us to change the grouping of terms without changing the value of the expression. For the expression \((7x + 8) + 20\), rearrange the brackets to group the constant terms together:\(7x + (8 + 20)\).

Step 3: Simplify the Inner Expression

Inside the parentheses, simplify the sum of the numbers:\(8 + 20 = 28\).Now, replace the inner expression with the sum:\(7x + 28\).

Step 4: Present the Simplified Expression

The simplified expression after applying the associative property and simplifying is:\(7x + 28\).

Key Concepts

Prealgebra Concepts

Prealgebra is a foundational branch of mathematics that introduces students to basic algebraic concepts. It forms the basis for understanding more advanced topics in algebra.
One crucial aspect of prealgebra is understanding how to work with algebraic expressions, which are mathematical phrases that can contain numbers, variables, and operations.
A key skill in prealgebra is learning how to manipulate these expressions to simplify them. This involves identifying parts of the expression and applying different mathematical properties to combine or reorganize the terms in beneficial ways.

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics, especially in prealgebra and algebra. The objective is to rewrite expressions in a simpler or more convenient form without changing their overall value.
In the given exercise, the expression \((7x + 8) + 20\) is simplified by applying the associative property. By changing the grouping, we can make mental calculations easier.

• First, it is important to look for opportunities where terms can be combined. In our example, combining the constant terms is a direct benefit.
• After rearranging using the associative property, perform basic arithmetic operations, such as addition or subtraction, to simplify.

The ultimate goal of simplifying an expression is to make it less complex and easier to understand or further manipulate in equations.

Algebraic Properties

Algebraic properties are a set of rules that guide how mathematical operations are performed and simplified. The associative property is one of these important rules.
The associative property is specifically applicable to addition and multiplication. It states that the way in which numbers are grouped does not affect the sum or product.

• In other words, for any numbers \(a, b,\) and \(c\), (\(a + b) + c = a + (b + c)\).
• This means you can freely rearrange the parentheses when adding or multiplying numbers, allowing flexibility in how you solve problems.

By using the associative property, you can simplify expressions and quickly find solutions, as seen in the example expression \((7x + 8) + 20 = 7x + (8 + 20) = 7x + 28\). This helps achieve a clearer and more concise final expression.