Problem 38 - Prealgebra

Question

Simplify each expression. $$(m+8)+4$$

Step-by-Step Solution

Answer

The simplified expression is \(m + 12\).

Step 1: Identify the expression

We need to simplify the expression \((m + 8) + 4\). This expression involves combining numbers and terms.

Step 2: Distribute the Addition

The expression \((m + 8) + 4\) involves adding 4 to the sum of \(m + 8\). Since addition is associative, we can simplify the expression by adding the numbers first.

Step 3: Add the Constants

Combine the numbers 8 and 4. Calculate \(8 + 4 = 12\).

Step 4: Write the Simplified Expression

Replace the sum of the constants back into the expression, resulting in \(m + 12\).

Key Concepts

Associative Property

The associative property is a fundamental concept in both arithmetic and algebra that simplifies many mathematical expressions. It states that the grouping of numbers in addition or multiplication does not affect their sum or product. This means when you're adding or multiplying multiple numbers, it doesn't matter how you group them, the result will be the same. Let's break this down with an example:

• Consider the expression \((a + b) + c\). According to the associative property, this is equivalent to \(a + (b + c)\).
• In terms of multiplication, \((a \times b) \times c\) is the same as \(a \times (b \times c)\).

In our exercise, we can use this property to simplify the expression \((m + 8) + 4\). Here, even though \(m\) is a variable, the constants can be combined together first due to the associative property. So, instead of adding \(4\) to \(m\) and then adding \(8\), it’s often easier to combine \(8 + 4\) first, which results in \(12\), and then add \(m\), yielding the simpler expression \(m + 12\). This property helps streamline calculations and can make solving problems quicker and more straightforward.

Combining Like Terms

Combining like terms is a critical skill in algebra that involves simplifying expressions by bringing together terms that have the same variable(s) to the same power. This is a method used to make expressions simpler and more concise. Understanding when and how to combine like terms is key to solving complex algebraic equations. Here's how it works:

• Identify terms with the same variables and the same exponents; for instance, \(3x\) and \(5x\) are like terms because they both have the variable \(x\).
• Constant terms (like numbers without variables) are also like terms, such as \(8\) and \(4\) in our example.
• Once identified, add or subtract coefficients (the numbers in front of the variables or constants).

In the original exercise, while \(m\) is a variable and cannot be combined with the constants \(8\) and \(4\), you can add the constants together, as they are like terms, resulting in \(12\). The simplified expression \(m + 12\) is cleaner and helps in further calculations.

Prealgebra

Prealgebra serves as the foundation for all future math studies, introducing students to the world of algebra in a way that builds confidence and understanding. It focuses on basic arithmetic operations along with the fundamental principles of algebra that are crucial for success in more advanced mathematics. Here’s what you can expect:

• Topics cover operations such as addition, subtraction, multiplication, and division, often introducing variables like \(x\) and \(m\).
• Understanding properties of operations, such as the commutative and associative properties, to simplify calculations.
• The concept of variables is also introduced, allowing expressions like \(m + 12\), which demonstrates an early form of algebraic thinking.

This stage is vital for helping students grasp the logic behind mathematical expressions and prepare them for algebra, where they will solve equations and work with inequalities. Prealgebra lays the groundwork by familiarizing students with combining like terms and utilizing properties such as the associative property. By mastering these concepts, students can move forward in their mathematical education with confidence and capability.