Problem 60 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$c(2 a+10 b)$$

Step-by-Step Solution

Answer

Question: Expand the following expression using the distributive property: $$c(2a + 10b)$$ Answer: $$2ac + 10bc$$

Step 1: Identify the terms to be distributed

In this case, the terms within the parentheses are \(2a\) and \(10b\). The constant value \(c\) will be distributed to these terms.

Step 2: Apply the distributive property

Distribute \(c\) to the two terms inside the parentheses: $$c(2a + 10b) = c(2a) + c(10b)$$

Step 3: Multiply the constant with each term

Multiply \(c\) to each term separately: $$c(2a) + c(10b) = 2ac + 10bc$$ Now, the expression has been expanded using the distributive property. The final solution is: $$c(2a + 10b) = 2ac + 10bc$$

Key Concepts

Algebraic Expansion

Algebraic expansion is a fundamental principle in mathematics that allows us to transform expressions into a form that is potentially easier to work with. Imagine you're unpacking a box filled with smaller boxes, each containing various items. Similarly, when expanding an algebraic expression, you're essentially opening up and distributing terms contained within parentheses to make the equation more usable.

Use algebraic expansion to solve more complex mathematical problems by breaking them down into simpler components. This process simplifies interactions between numbers and variables, making it particularly valuable for solving equations, evaluating expressions, and performing calculus operations.

In our example, using algebraic expansion on the expression \(c(2a + 10b)\) involves distributing the constant \(c\) across each term inside the parentheses. This effectively "expands" the expression to make calculations straightforward.

Multiplication of Terms

Understanding the multiplication of terms is key in the process of algebraic expansion. Remember, when you multiply terms together, you are essentially applying the distributive property, which allows each term in one set of expressions to interact with every term in another.

In the provided example, multiplying \(c\) across the terms \(2a\) and \(10b\) involves individually combining \(c\) with \(2a\) and then separately with \(10b\). This is how distributive property works.

• Multiply the constant or the coefficient \(c\) with \(2a\) resulting in \(2ac\).
• Then multiply \(c\) with \(10b\), giving us \(10bc\).

This process demonstrates how terms multiply to form new expressions, incorporating coefficients, variables, and constants into unified terms that describe relationships within an equation.

Mathematical Expressions

Mathematical expressions are a combination of numbers, variables, and operations that together represent a specific value or a simplified form of a problem. They act as the language of algebra, allowing you to convey complex calculations succinctly.

When you look at an expression like \(c(2a + 10b)\), you're seeing a compact form of a relationship that can be expanded for clarity, as demonstrated in earlier steps. A mathematical expression can range in complexity from simple sums to intricate polynomial equations. Understanding and manipulating these expressions is a foundational skill in algebra.

Recognizing how expressions can be expanded and simplified is essential for solving equations, interpreting graphs, and performing mathematical operations. By learning to manage expressions effectively, you gain the tools needed to advance through more complex topics in mathematics with ease.