Problem 343 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 8(4 y+9) $$

Step-by-Step Solution

Answer

32y + 72

Step 1: Identify the Distributive Property

The Distributive Property states that for any numbers or variables, \[a(b + c) = ab + ac\]. This property will be used to distribute the 8 to both terms inside the parentheses.

Step 2: Apply the Distributive Property

Distribute the 8 to both terms inside the parentheses: \[8(4y + 9) = 8 \times 4y + 8 \times 9\].

Step 3: Multiply the Numbers

Calculate the products: \[8 \times 4y = 32y\] and \[8 \times 9 = 72\].

Step 4: Combine the Terms

Add the two results together: \[32y + 72\].

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a key concept in algebra that helps in making complex equations more manageable. It involves reducing expressions to their simplest form. To simplify an expression, we perform operations such as distribution, combining like terms, and more. This process makes solving equations easier. For instance, consider the expression given in the exercise: \(8(4y + 9)\). By simplifying it, we can turn a complex problem into something much more straightforward, like \(32y + 72\). Remember, the goal is to make the expression as neat and concise as possible while preserving its original value.

Multiplication in Algebra

Multiplication in algebra is not much different from regular multiplication, but it involves variables and constants. Whenever you see a term like \(8(4y + 9)\), you know you need to apply multiplication. The key property to use here is the Distributive Property. This property allows you to multiply each term inside the parenthesis by the term outside. So, \(8(4y + 9)\) becomes \8 \times 4y \ and \8 \times 9\. Hence the multiplication steps will yield \(32y + 72\).

Combining Like Terms

Combining like terms is another fundamental skill in algebra. It involves merging terms that have the same variable raised to the same power. For example, in the expression \(32y + 72\), we can't combine \(32y\) and \(72\) because they are unlike terms—one has a variable \(y\) and the other is a constant. However, if you had \(32y + 8y \), you could combine them to get \(40y\). This step simplifies the expression and often makes it easier to solve equations later on. Look for terms with the same variable to combine them efficiently.