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Chapter 1: Problem 55

Rewrite each expression using the distributive property. Simplify if possible. $$(8-2) 4$$

Short Answer

Expert verified
The expression \((8-2)4\) can be rewritten using the distributive property as \(4(8) - 4(2)\), which simplifies to \(32 - 8\), resulting in a final answer of \(24\).

Step by step solution

01

Identify the distributive property

The distributive property states that the product of a number and a sum is equal to the sum of the products of the number with each term in the sum. In other words, for any numbers a, b, and c: \(a(b + c) = ab + ac\) In our given expression, we have: \((8 - 2) 4\) Notice that we can apply the distributive property here by treating \(4\) as the "a" term and the \((8 - 2)\) as the sum of "b" and "c" terms.
02

Apply the distributive property

Now, we can apply the distributive property to our expression: \((8 - 2) 4 = 4(8) - 4(2)\)
03

Perform arithmetic operations

Finally, we can perform the arithmetic operations to simplify the expression: \(4(8) - 4(2) = 32 - 8\) \(32 - 8 = 24\) So, after rewriting the expression using the distributive property and simplifying it, we get: \((8-2) 4 = 24\)

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Key Concepts

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

Simplifying Expressions
When we talk about simplifying expressions, we mean turning them into their simplest or most compact form. The main goal is to get the expression into a version that is easy to work with or understand. Think of it like cleaning your desk 鈥� you want everything to be organized and neat!

In mathematical expressions, this involves:
  • Combining like terms
  • Performing any addition, subtraction, multiplication, or division as needed
  • Applying properties like the distributive property to rearrange or reduce the expression
Let's consider the expression \((8-2) 4\). First, we use the distributive property to break it down, and then we solve the arithmetic part: \(4 \times 8 - 4 \times 2\). After calculating, we find it simplifies to \(24\). This step-by-step approach makes even complex expressions easy to handle.
Arithmetic Operations
Arithmetic operations include basic mathematical actions such as addition, subtraction, multiplication, and division. Mastering these operations is crucial because they form the foundation of much more complex math concepts.

In our example with \((8-2) 4\), we used multiplication and subtraction, both key arithmetic operations:
  • Multiply first: \(4 \times 8\) and \(4 \times 2\)
  • Then subtract: \(32 - 8\)
  • Finally, we reach the result: \(24\)
Each step is like a piece of a puzzle. Together they build towards the complete picture or solution. It's vital to follow the order of operations (often remembered by PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to ensure accuracy.
Algebraic Expressions
Algebraic expressions might seem intimidating, but they are simply a combination of numbers, variables, and operations. They represent things that can change or aren鈥檛 fixed in value.

One key to working with algebraic expressions is understanding the rules and properties, like the distributive property. This property helps to manipulate expressions, making it easier to solve them. In our example:
  • The expression \((8-2) 4\) doesn't have variables, making it a numerical expression.
  • The rules we use are the same as those for algebraic expressions; we break it down using distributive property to simplify: \(4(8) - 4(2)\).
By practicing these steps, you'll be able to handle any algebraic expressions that come your way. Remember, it's all about understanding the parts and how they come together.

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