Problem 65 - Prealgebra

Question

Simplify each expression. $$-8(r-5)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-8r + 40\).

Step 1: Distribute the Negative Sign

Start by distributing the negative sign across the parenthesis. This means multiplying -8 with each term inside the parenthesis. The expression is \(-8(r - 5)\). Apply the distribution: \(-8 \times r = -8r\) and \(-8 \times (-5) = +40\).

Step 2: Simplify the Expression

Combine the terms obtained from distribution. The terms after distribution are \(-8r + 40\). We write this as the simplified expression.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept that helps to simplify expressions. It involves multiplying a single term by each term within a set of parentheses. For the expression \[-8(r - 5)\], we apply the distributive property by multiplying -8 with \(r\) and -5 .

• Negative times positive (\(-8 \times r\)) equals \(-8r\).
• Negative times negative (\(-8 \times -5\)) equals \(+40\), since two negatives make a positive.

Using the distributive property allows us to break down complex expressions into smaller, more manageable parts, which can then be simplified further.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. In the expression \(-8r + 40\),

Constants are fixed values like \(40\), which do not change.

Variables, like \(r\), are symbols that represent numbers and can vary.

Solving algebraic expressions involves using mathematical operations like addition, subtraction, and distribution to make expressions simpler and easier to understand.

Negative Numbers

Negative numbers are numbers less than zero and they play a crucial role in algebra, especially when multiplied within expressions. When you multiply two negative numbers, like in our expression \(-8 \times -5\), the product is positive because the negatives cancel each other out.

• Negative times positive results in a negative, such as \(-8 \times r = -8r\).
• Negative times negative results in a positive, such as \(-8 \times -5 = +40\).

Understanding how negative numbers interact is critical in simplifying expressions and solving equations in algebra. It helps ensure computations are executed correctly and results are accurate.