Problem 73 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -8(r+3) $$

Step-by-Step Solution

Answer

-8r - 24

Step 1: Identify the Distributive Property

The distributive property states that for any numbers or expressions a, b, and c, the expression \(a(b + c)\) can be rewritten as \(ab + ac\). In this case, \(-8(r + 3)\) corresponds to \(a = -8\), \(b = r\), and \(c = 3\).

Step 2: Distribute the -8

To apply the distributive property, multiply -8 by each term inside the parentheses. This gives: \(-8 \times r + (-8) \times 3\).

Step 3: Multiply -8 by the Terms Inside the Parentheses

Multiply \(-8 \times r \) to get \(-8r\) and \(-8 \times 3\) to get \(-24\).

Step 4: Combine the Results

Combine the two products to get the final expression, which is \(-8r - 24\).

Key Concepts

algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In algebra, we often use letters to represent unknown values. This allows us to form equations and expressions that model real-world situations.

One key aspect of algebra is understanding and using properties like the distributive property. By mastering these properties, we can simplify complex expressions and solve equations more easily.

In the given exercise, we're asked to use the distributive property to simplify the expression \(-8(r+3)\). This is a foundational skill that helps us work with and understand more advanced algebraic concepts.

simplifying expressions

Simplifying expressions is a crucial skill in algebra. It involves reducing an expression to its simplest form by performing operations and combining like terms.

In the exercise, we start with the expression \(-8(r+3)\). To simplify it, we use the distributive property, which allows us to break apart the expression inside the parentheses and multiply each term by \-8\.

Here's how the steps look:

• First, distribute \-8\ to each term inside the parentheses: \-8 \times r\ and \-8 \times 3\.
• Next, perform the multiplication to get \-8r\ and \-24\.
• Finally, combine these results to achieve the simplified expression: \-8r - 24\.

By following these steps, we simplify the original expression and make it more manageable for further calculations.

multiplication

Multiplication is one of the basic operations in mathematics and plays a significant role in algebra. When using the distributive property, multiplication allows us to combine and simplify expressions.

In our exercise, we need to multiply \-8\ by each term inside the parentheses.

• Begin with \-8 \times r\:
\-8r\ is the result because multiplying a number by a variable involves just multiplying the coefficients together.
• Next, do \-8 \times 3\:
This gives us \-24\ because we are multiplying two integers.

By performing these multiplications, we break down the expression \(-8(r+3)\) into components that are easier to handle.

Mastering multiplication is essential for algebraic manipulations and for applying properties like the distributive property effectively.