Problem 344 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 9(3 w+7) $$

Step-by-Step Solution

Answer

\(27w + 63\).

Step 1: Identify the terms

Identify the terms inside the parentheses and the term outside of the parentheses. Here, the terms inside the parentheses are \(3w+7\), and the term outside the parentheses is \(9\).

Step 2: Apply the Distributive Property

Use the Distributive Property, which states that \(a(b+c) = ab + ac\). In this problem, apply the property by multiplying \(9\) by each term inside the parentheses: \(9 \times 3w + 9 \times 7\).

Step 3: Multiply each term

Carry out the multiplications: \(9 \times 3w = 27w\) and \(9 \times 7 = 63\).

Step 4: Write the simplified expression

Combine the results from Step 3 to write the simplified expression: \(27w + 63\).

Key Concepts

simplifying expressions

Simplifying expressions in algebra involves reducing them to their most concise and manageable form. This helps make complex equations easier to work with and understand. In the given exercise, we're tasked with simplifying the expression using the Distributive Property. Starting with the equation $$9(3w + 7)$$, our goal is to rewrite it in a simpler form.

To do this, we follow a structured approach:

• Identify the terms inside and outside the parentheses.
• Apply the Distributive Property.
• Multiply each term.
• Combine the results.

This process helps in making sure that all parts of the expression are accounted for and properly simplified.

algebraic multiplication

Algebraic multiplication is a core concept in algebra that involves multiplying numbers and variables. In our example, we use the Distributive Property to handle algebraic multiplication:

Applying the Distributive Property means we multiply the number outside the parentheses (which is 9) by each term inside the parentheses ( \(3w \text{and} 7\) ). In this case, we perform the following steps:

• First, multiply 9 by 3w to get 27w.
• Next, multiply 9 by 7 to get 63.

The results from these two multiplication operations, 27w and 63, are then combined to form the simplified expression: \(27w + 63\).

intermediate algebra

Intermediate algebra builds upon fundamental algebra concepts and introduces more complex operations and properties. The Distributive Property is one of these important properties.

In this exercise, we used the Distributive Property to simplify the expression. This involves recognizing patterns and applying rules systematically. For instance, in the expression 9(3w + 7), the Distributive Property ensures that the term outside the parentheses, 9, is distributed to both terms inside the parentheses: 3w and 7.

The ability to simplify expressions using algebraic rules is a crucial skill in intermediate algebra as it prepares you for more advanced topics and helps in solving more complex equations. By mastering these steps, you develop a stronger foundation in algebraic manipulation.