Problem 8 - Prealgebra

Question

Use the distributive property to combine each of the following pairs of similar terms. $$8(x-2)$$

Step-by-Step Solution

Answer

The expression simplifies to \(8x - 16\).

Step 1: Identify Components Inside Parentheses

Look at the expression inside the parentheses: \( (x - 2) \). These are the terms that will be multiplied by 8.

Step 2: Apply Distributive Property

The distributive property states \( a(b + c) = ab + ac \). In this case, replace \( a \) with 8, \( b \) with \( x \), and \( c \) with \(-2\). This gives \( 8 \cdot x + 8 \cdot (-2) \).

Step 3: Multiply Each Term

Calculate \( 8 \cdot x = 8x \) and \( 8 \cdot (-2) = -16 \).

Step 4: Combine Results

Combine the results of the multiplication: \( 8x - 16 \). This expression is the simplified form of the original expression using the distributive property.

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental component in algebra. They consist of numbers, variables, and arithmetic operations. For instance, the expression \(8(x-2)\) can be broken down into different parts:

• 8 is a coefficient, which indicates how many times a value will be multiplied.
• \(x\) is a variable, acting as a placeholder for unknown values.
• \(-2\) is a constant term, which remains fixed.
• The parentheses show terms that need combining or distributing.

These elements come together to form what are known as algebraic expressions. They are key in representing real world scenarios mathematically. Understanding how to manipulate and solve these expressions can help in areas like geometry, physics, and even computer science. They allow us to describe patterns and predict outcomes in a structured way.
Recognizing each part of an algebraic expression is the first step in solving or simplifying it. Thus, whenever you come across an expression, identifying its components will guide your approach in solving it effectively.

Multiplication of Terms

The multiplication of terms inside an expression often uses the distributive property. It’s a cornerstone in algebra, simplifying calculations and restructuring expressions. When multiplying a number by a group of terms in parentheses, every term inside must be multiplied individually by the number outside.
Consider the expression \(8(x-2)\):

• First, recognize that you need to multiply 8 by each term inside the parentheses.
• Multiplying 8 by \(x\) results in \(8x\).
• Subsequently, multiplying 8 by -2 produces -16.

Using this method ensures that you distribute, or apply, the multiplication evenly across all terms within the parentheses. It eliminates errors caused by missing out on terms, providing a complete and accurate expression as a result.
Multiplying terms in algebra requires attention to detail, particularly with signs and variables involved. Consistency in applying the distributive property leads to reliable and precise outcomes.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. This often means completing operations such as distribution, combination of like terms, and even factorization. Simplification makes expressions neater and more understandable. In our example, \(8(x-2)\), simplifying begins with the distributive step:

• Use the distributive property to multiply each term within the parentheses.
• You get \(8x - 16\), which is simpler than the original form.

Sometimes, simplification might not stop at distribution. If there are like terms, they should be combined to further tidy the expression. For example, if the resulting expression was \(8x - 16 + 4x\), you would then combine \(8x\) and \(4x\) to get \(12x - 16\).
Remember, simplifying expressions is an ongoing process, requiring you to constantly check if it can be condensed further. The goal is to make the expression as concise as possible while retaining the same value.