Problem 20 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expression without parentheses is \(8a - 8b\).

Step 1: Understand the expression

The given expression is \((a-b)8\). Here, 8 is multiplying the difference \((a-b)\). We will use the distributive property to remove the parentheses.

Step 2: Apply the Distributive Property

The distributive property states that \(a(b + c) = ab + ac\). In our expression, we can write it as \((8a - 8b)\), by distributing the 8 to both \(a\) and \(-b\).

Step 3: Simplify the Expression

After distribution, check if any further simplification is needed. Here, the expression \(8a - 8b\) cannot be simplified further.

Key Concepts

Properties of Real Numbers

The properties of real numbers are fundamental rules that govern arithmetic operations like addition, subtraction, multiplication, and division. These properties help us manipulate and simplify expressions and equations efficiently and accurately. Some key properties include:

• Commutative Property: This property states that the order of numbers does not affect their sum or product. For example, in addition, \(a + b = b + a\) and in multiplication, \(ab = ba\).
• Associative Property: This suggests that how numbers are grouped doesn't change their sum or product. For instance, \((a + b) + c = a + (b + c)\) for addition.
• Distributive Property: This is crucial when working with expressions that involve parentheses. It allows us to distribute a multiplier over terms inside parentheses: \(a(b + c) = ab + ac\), which is particularly helpful for simplifying expressions.

These properties are the backbone of algebra and are used extensively to simplify and manipulate mathematical expressions.

Simplification

Simplification in algebra involves reducing expressions to their simplest form without changing their value. This process often involves distributing terms, combining like terms, and reducing fractions. The goal is to make expressions easier to understand and work with. In our exercise, simplification started with the application of the distributive property on the expression \((a-b)8\) to remove the parentheses, resulting in \(8a - 8b\). After using the distributive property, you should always check for like terms—terms that have the same variable raised to the same power. If there are any, combine them to simplify further. In our exercise, however, once we distributed, there were no like terms to combine. Hence, \(8a - 8b\) is already in its simplest form.

Multiplication in Algebra

Multiplication in algebra just like in arithmetic involves combining quantities. But in algebra, it often involves variables alongside numbers.Algebraic multiplication relies heavily on the properties of multiplication, like the distributive property, especially when dealing with terms inside parentheses. When you encounter an expression like \((a-b)8\), multiplication is used to distribute the 8 across the terms inside the parenthesis.Here’s how it works:

• You multiply 8 by each term inside the parentheses separately: \(8 \times a = 8a\) and \(8 \times -b = -8b\).
• Combine the results into one expression: \(8a - 8b\).

This step-by-step process not only simplifies the expression but also illustrates the practical use of the distributive property in algebraic multiplication.