Problem 360 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ -9(9 a+4) $$

Step-by-Step Solution

Answer

-81a - 36

Step 1: Identify the Distributive Property

The Distributive Property states that for any real numbers a, b, and c, the equation \[ a(b + c) = ab + ac \] holds true.

Step 2: Apply the Distributive Property

Use the Distributive Property to multiply \(-9\) by each term inside the parentheses \( (9a + 4) \) separately. This gives: \(-9 \times 9a + (-9) \times 4\)

Step 3: Simplify Each Term

Carry out the multiplication for each term: \[ -9 \times 9a = -81a \] \[ -9 \times 4 = -36 \]

Step 4: Combine the Results

Combine the simplified terms to get: \[ -81a - 36 \]

Key Concepts

Simplifying Expressions

Simplifying expressions means making them easier to understand or solve. In algebra, we often do this to solve equations more quickly. Simplifying can involve combining like terms, using the distributive property, or performing basic arithmetic operations. For example, in the exercise \ \[-9 (9a + 4)\], we use the Distributive Property to simplify it. This makes the expression less complex and shows the relationship between terms more clearly. Breaking down the expression step-by-step helps to avoid errors and leads to accurate results.

Multiplication

Multiplication is one of the basic arithmetic operations and is vital for simplifying expressions. When you multiply, you are essentially adding the same number multiple times. In our example, \(-9 (9a + 4)\), we multiply -9 by each term inside the parentheses. \(-9 \times 9a+(-9) \times 4\) is the operation we perform. This step-by-step approach helps in understanding how multiplication distributes over addition within parentheses. Multiplication can get more complex as more variables and terms are involved, but the foundational principles stay the same.

Algebraic Terms

Algebraic terms are the building blocks of algebraic expressions. They include numbers, variables, and the products of numbers and variables. Each term in an expression is separated by a plus (+) or minus (−) sign. In our exercise, \(9a + 4\) consists of two algebraic terms: \(9a\) and \(4\). When we simplify using the distributive property, each term is multiplied individually by -9. Understanding how to manipulate algebraic terms is critical for simplifying more complex expressions.

Intermediate Algebra

Intermediate algebra builds upon basic algebra concepts by introducing more complex operations and properties. Using the Distributive Property, like in our example, is a key skill. Instead of just performing basic operations, you learn to apply properties like distribution, combine like terms, and solve multi-step problems. In the exercise \(-9(9a + 4)\), we used intermediate algebra techniques to simplify the expression to \(-81a - 36\). Mastering these skills is crucial for tackling higher-level math courses and real-world problem-solving scenarios.