Problem 70 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities.

Step-by-Step Solution

Answer

Question: Expand the expression \(a(b+3c)\) using the distributive property. Answer: \(ab + 3ac\)

Step 1: Identify the distributive elements

Here, the \(a\) is being multiplied by the entire \((b+3c)\) term. So, we need to apply the distributive property to expand the expression.

Step 2: Apply the distributive property

To apply the distributive property, multiply the outside term \(a\) by each term inside the parenthesis: \(ab + a(3c)\).

Step 3: Simplify the expression

Now, we need to simplify the expression by multiplying the remaining terms: \(a(3c)\) becomes \(3ac\). Therefore, the expanded expression is: $$ a(b+3c) = ab + 3ac $$

Key Concepts

Algebraic Expressions

Algebraic expressions are the building blocks of algebra, a branch of mathematics that uses symbols and letters to represent numbers and operations. These expressions can contain variables, constants, and mathematical operators such as addition, subtraction, multiplication, and division. For example, in the expression \(a(b+3c)\), \(a\), \(b\), and \(c\) represent variables, which means they can take on any value.

Algebraic expressions do not have an equality sign (unlike equations). They are often used to represent general forms of relationships in mathematics, like formulas or equations. Working with algebraic expressions includes tasks such as simplifying, factoring, and expanding expressions to make them easier to interpret and solve.

Understanding algebraic expressions means recognizing and effectively manipulating these components to understand mathematical relationships. By grasping this concept, learners can build their problem-solving skills and work towards more complex mathematical problems.

Expanding Expressions

Expanding expressions is a key concept that involves transforming a compact algebraic expression into a more detailed or extended form. This process is often achieved using properties like the distributive property, which allows us to remove parentheses by multiplying each term inside them by the term outside.

When you expand an expression, you essentially "distribute" the multiplication over addition or subtraction within parentheses. For example, consider the expression \(a(b+3c)\). To expand this expression, you multiply the outside term \(a\) by each term inside the parentheses, resulting in \(ab + 3ac\).

Expanding expressions helps in simplifying complex algebraic equations and enables easier manipulation when solving problems. It's a crucial step towards solving equations, making it fundamental for students to understand and practice.

Multiplication in Algebra

Multiplication in algebra follows similar rules to multiplication with numbers but involves additional considerations because of variables. It is a foundational operation that combines terms through multiplication. For algebraic expressions, multiplication often involves distributing a term across terms within parentheses or combining like terms.

In the expression \(a(b + 3c)\), multiplication is applied by using the distributive property to spread \(a\) across the terms \(b\) and \(3c\). This action results in the terms \(ab\) and \(3ac\), showcasing how multiplication interacts with different terms in an expression.

Understanding multiplication in algebra is important because this operation is pivotal in expanding expressions, simplifying equations, and solving for unknowns. Additionally, mastering multiplication in algebra helps in dealing with polynomials and other higher-level mathematics, where these concepts are frequently applied.