Problem 106 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 12 n(8) $$

Step-by-Step Solution

Answer

The simplified form is \(96n\).

Step 1: Identify Constants and Variables

In the expression \(12n(8)\), we have a constant (12), a variable \(n\), and another constant (8). The expression represents multiplication between these components.

Step 2: Rearrange the Multiplication

Since multiplication is commutative, we can rearrange the expression without changing its value. This means \(12n(8)\) can be rearranged as \(12 \times 8 \times n\).

Step 3: Perform Constant Multiplication

Now multiply the constants: \(12\) and \(8\). Doing this gives us \(96\). So, \(12 \times 8 = 96\).

Step 4: Simplify the Expression

Substitute back the multiplied result into the expression to get \(96 \times n\). So, the simplified form of the original expression is \(96n\).

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is all about making them as compact and easy to work with as possible. It's the process of condensing an expression to its simplest form while retaining its original meaning. This often involves combining like terms and performing basic arithmetic operations.

The goal of simplifying is typically to make calculations easier and to provide a clear view of the relationship between different parts of an expression. For example, in the expression \(12n(8)\), we simplify it to become \(96n\). This makes it more straightforward to understand and work with.

Here are a few tips to remember when simplifying expressions:

• Always perform arithmetic with any constants or coefficients first.
• Look for like terms, and consolidate them if possible.
• Apply operations carefully, ensuring that the original expression's value doesn't change.
• Consider properties of numbers like commutative or distributive properties to make your task easier.

Variable Manipulation

Variable manipulation involves adjusting, rearranging, or using variables in algebraic expressions in a way that helps simplify or solve equations. When dealing with expressions like \(12n(8)\), it's crucial to treat variables with special care to ensure that the expression remains accurate.

Variables are symbols (often letters) that represent numbers or values in an expression. The key to manipulating variables effectively includes a few basic rules:

• Keep track of the variables and where they are in the expression.
• Remember that variables can be multiplied and divided just like numbers, but take caution when adding or subtracting, as they must be like terms to be combined.
• Apply the order of operations (PEMDAS/BODMAS) to make sure all calculations are done in the correct order.
• Respect the properties of variables; they cannot be switched around unless you're using a property that allows it, like the commutative property for multiplication.

Commutative Property

The commutative property is a basic but very helpful mathematical principle, especially when working with algebraic expressions. This property states that the order in which two numbers are added or multiplied does not change their result.

For addition, it means \(a + b = b + a\), and for multiplication, it's \(a \times b = b \times a\). This property allows flexibility in rearranging an algebraic expression like \(12n(8)\) to \(12 \times 8 \times n\), making it easier to perform calculations in steps.

Some important aspects of the commutative property include:

• It applies only to addition and multiplication, not to subtraction or division.
• Utilizing this property can simplify complex expressions and make mental calculations simpler.
• It's foundational, so understanding it can help in more complicated topics later in algebra.
• Remember that rearranging terms can often show useful patterns or make further simplifications possible.