Problem 24 - Algebra and Trigonometry

Question

Use properties of real numbers to write the expression without parentheses. \((a-b) 8\)

Step-by-Step Solution

Answer

The expression simplifies to \(8a - 8b\).

Step 1: Identify the Property Needed

In this expression, we need to distribute the number 8 to each term inside the parentheses. The property that allows us to do this is called the Distributive Property of multiplication over subtraction.

Step 2: Apply the Distributive Property

The Distributive Property states that for any real numbers, \(a\), \(b\), and \(c\), \(c(a - b) = c \cdot a - c \cdot b\). Applying this to the expression \((a-b)8\), we get: \(8 \cdot a - 8 \cdot b\).

Step 3: Simplify the Expression

Perform the multiplication to simplify the expression. The final expression is: \(8a - 8b\).

Key Concepts

Understanding Real Numbers

Real numbers are fundamental elements in mathematics, including all the numbers that you can think of. They consist of decimal numbers, integers, fractions, and even irrational numbers like \( \pi \) and \( \sqrt{2} \). Real numbers can be represented on the number line, encompassing both positive and negative values, as well as zero.
Real numbers are crucial when dealing with algebraic expressions. They provide a comprehensive system that allows us to perform various operations such as addition, subtraction, multiplication, and division.
Whether you're handling large computations or simple arithmetic, understanding real numbers is essential. They bridge the gap between different types of numbers and help in their seamless manipulation in equations and expressions.

Multiplying with the Distributive Property

Multiplication is one of the basic arithmetic operations involving increasing or scaling a number by another number. In an expression like \((a - b) \cdot 8\), the distribution of multiplication plays a pivotal role. This is where the Distributive Property comes into play.
The Distributive Property of multiplication over subtraction allows us to simplify expressions by distributing the multiplication across terms within parentheses. It states that \(c(a-b) = c \cdot a - c \cdot b\).

• First, multiply 8 by \(a\), resulting in \(8 \cdot a\).
• Then, multiply 8 by \(b\), resulting in \(8 \cdot b\).

Thus, the expression \((a-b) \cdot 8\) is simplified to \(8a - 8b\) using the distributive property.

Simplifying Expressions

Simplification is the process of making an expression easier to work with by combining like terms and performing arithmetic operations. Once we distribute the multiplication across the terms of the expression as shown above, simplifying helps us to tidy up the expression.
For the expression \(8a - 8b\), simplifying involves merely completing the multiplication and writing the expression clearly without any unnecessary parentheses or operations.

• Double-check to ensure there are no further operations needed, like combining like terms.
• Present the expression in its simplest form, which in this case is \(8a - 8b\).

This step is crucial for making further calculations straightforward and error-free. Simplified expressions provide clarity and reduce chances for mistakes in subsequent equations.