Problem 42 - Prealgebra

Question

Use the Distributive Property to rewriteh expression as an equivalent algebraic expression. \(4(x-8)\)

Step-by-Step Solution

Answer

The equivalent expression is \(4x - 32\).

Step 1: Understanding the Distributive Property

The Distributive Property in algebra allows you to multiply a single term by each term inside a parenthesis. It is expressed as \(a(b + c) = ab + ac\). Apply this property to simplify algebraic expressions.

Step 2: Identify Parts for Distribution

In the given expression \(4(x - 8)\), identify the term outside the parentheses, which is \(4\), and the terms inside the parentheses, which are \(x\) and \(-8\). These will be the terms to which the Distributive Property is applied.

Step 3: Apply the Distributive Property

Multiply the term outside the parenthesis, \(4\), by each of the terms inside the parenthesis, \(x\) and \(-8\). This gives: \(4 \cdot x\) and \(4 \cdot (-8)\), which simplifies to \(4x\) and \(-32\), respectively.

Step 4: Write the Equivalent Expression

Combine the results from distributing \(4\) across \(x\) and \(-8\) to form the equivalent expression. The resulting expression from the distribution is \(4x - 32\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They are the building blocks in algebra that allow us to express mathematical ideas in a compact form. For example, in the expression \(4(x-8)\), we have:

• An external multiplier: 4
• A variable: \(x\)
• A constant: \(-8\)
• A subtraction operation inside the parentheses: \(x - 8\)

Algebraic expressions can represent real-world problems and can be manipulated using various mathematical rules and properties. One such powerful tool is the Distributive Property, which we commonly use to simplify and rewrite these expressions.

Simplifying Expressions

Simplifying expressions in algebra involves rewriting them in a more straightforward or reduced form without changing their value. This process often makes the expression easier to work with when solving equations or performing further operations.
When simplifying, we need to apply mathematical properties like the Distributive Property effectively. In our exercise, we have the expression \(4(x-8)\). Applying the Distributive Property, we multiply \(4\) by each term inside the parenthesis:

• Multiply \(4\) by \(x\), resulting in \(4x\)
• Multiply \(4\) by \(-8\), resulting in \(-32\)

Combine these results, and we have the simplified expression: \(4x - 32\).
Simplification helps in making complex expressions easier to interpret and solve.

Mathematical Operations

Mathematical operations are fundamental actions we perform on numbers and variables. These include addition, subtraction, multiplication, and division.
In algebra, multiplication is especially important when working with the Distributive Property. It allows us to "distribute" a multiplier across terms in parentheses. This operation is a crucial step for transforming expressions and is evident in simplifying \(4(x-8)\):

• The term \(4\) is multiplied across each term in \(x - 8\)
• Perform multiplication: \(4 \times x = 4x\) and \(4 \times -8 = -32\)

Each operation plays a vital role in reshaping expressions to solve them accurately or to make them ready for further analysis.