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Chapter 7: Problem 90

In the following exercises, simplify using the distributive property. $$4(x+8)$$

Short Answer

Expert verified
The simplified expression is \(4x + 32\).

Step by step solution

01

Identify the Expression

Look at the expression given: \(4(x+8)\). This expression has a number outside the parentheses that needs to be distributed to each term inside the parentheses.
02

Distribute the Number

Use the distributive property, which states that \(a(b + c) = ab + ac\). Here, \(4\) is the number outside the parentheses (a), and the terms inside the parentheses (b and c) are \(x\) and \(8\). Distribute \(4\) to both \(x\) and \(8\). Distribute \(4\) to \(x\): \(4 \times x\) equals \(4x\). Distribute \(4\) to \(8\): \( 4 \times 8\) equals \(32\).
03

Combine the Results

Combine the distributed terms to get the simplified expression. \(4x + 32\). This is the final simplified expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying expressions
Simplifying expressions is a fundamental skill in algebra. To simplify means to make an expression easier to handle, usually by performing all possible operations and reducing it to its simplest form. This often involves:

  • Combining like terms
  • Using mathematical properties such as the distributive property
In our original exercise, we started with the expression \(4(x+8)\). By applying the distributive property, we simplified it to \(4x + 32\). The simplification process made the expression cleaner and more manageable, which is often necessary before solving equations or further manipulating expressions in algebra.
algebraic expressions
Algebraic expressions consist of variables, numbers, and operations (like addition and multiplication). They are not equations鈥攖hey do not have an equals sign鈥攁nd they are used to represent mathematical relationships and functions. For example, in the expression \(4(x+8)\), we have:

  • A coefficient (4)
  • A variable (x)
  • A constant (8)
Understanding how to manipulate and simplify these expressions is crucial in algebra. Our goal is often to make these expressions as simple as possible, as seen when we transformed \(4(x+8)\) into \(4x + 32\). This step is vital because it lays the foundation for solving more complex problems.
prealgebra
Prealgebra is the stepping stone to higher-level math, focusing on the basic concepts of arithmetic and introducing algebraic thinking. Key topics include:

  • Basic arithmetic operations (addition, subtraction, multiplication, division)
  • Understanding variables and unknowns
  • Applying properties such as the distributive property
In our example, the exercise involves the distributive property鈥攁 concept often covered in prealgebra. This property helps us distribute a multiplier across terms within parentheses, making complex expressions simpler. For instance, transforming \(4(x+8)\) into \(4x + 32\) prepares students for the type of thinking and problem-solving they'll encounter in more advanced math courses. Understanding these foundational concepts is essential for progressing in mathematics.

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