Problem 5 - Prealgebra

Question

Combine by applying the distributive property. Assume all variables represent positive numbers. $$9 \sqrt{x}+3 \sqrt{x}-5 \sqrt{x}$$

Step-by-Step Solution

Answer

The combined expression is \( 7 \sqrt{x} \).

Step 1: Identify Common Factors

Notice that each term in the expression contains the common factor \( \sqrt{x} \). The expression is: \( 9 \sqrt{x} + 3 \sqrt{x} - 5 \sqrt{x} \).

Step 2: Apply the Distributive Property

Factor out the common factor \( \sqrt{x} \) from each term. This gives us: \( \sqrt{x}(9 + 3 - 5) \).

Step 3: Simplify Inside the Parentheses

Perform the arithmetic operation inside the parentheses: \( 9 + 3 - 5 = 7 \). This simplifies the expression to \( \sqrt{x} \cdot 7 \).

Step 4: Final Expression

Rewrite the expression by multiplying the factor \( \sqrt{x} \) with the simplified result from the parentheses: \( 7 \sqrt{x} \).

Key Concepts

Understanding Common Factors

Common factors play a crucial role in simplifying expressions, especially when dealing with terms that include similar components. In mathematical terms, a common factor is a value or expression that is shared amongst all terms in an equation or expression. Identifying common factors is an essential step in making expressions more manageable and easier to solve.

When looking at an expression like \(9 \sqrt{x} + 3 \sqrt{x} - 5 \sqrt{x}\), you might notice that each term features the radical \(\sqrt{x}\). This radical is what we call a common factor because it appears in every term.

By recognizing the common factor, you can apply properties such as the distributive property effectively, paving the way for simplifying the whole expression effortlessly. Recognizing common factors not only makes the process smoother but also ensures accuracy by combining similar elements in a consistent, organized manner.

Simplifying Expressions

Simplifying expressions involves reducing the complexity of a mathematical statement while ensuring its value remains unchanged. It's about breaking down and combining like terms to create a more straightforward equation or result.

The process of simplifying expressions often involves applying the distributive property. The distributive property states that a common factor outside parentheses can be distributed across terms inside the parentheses. In our example, the expression \(9 \sqrt{x} + 3 \sqrt{x} - 5 \sqrt{x}\) simplifies by factoring out \(\sqrt{x}\) to become \(\sqrt{x}(9 + 3 - 5)\).

Next, perform the arithmetic within the parentheses: \(9 + 3 - 5\), which simplifies to \(7\). This further reduces the original expression to \(\sqrt{x} \cdot 7\) or \(7 \sqrt{x}\). Breaking down expressions into simpler forms allows for easier manipulation, which is particularly helpful when solving equations or evaluating functions further.

Introduction to Radicals

Radicals, often represented by the square root symbol \(\sqrt{}\), can be a complex topic for many learners. A radical expression usually involves a square root, cube root, or any higher-order root. In expressions, radicals allow us to work with numbers that aren't perfect squares, making them versatile in various calculations.

In the context of the expression \(9 \sqrt{x} + 3 \sqrt{x} - 5 \sqrt{x}\), the \(\sqrt{x}\) is a radical term. Radicals can sometimes make expressions seem daunting, but when you notice that each term shares a radical, it highlights the use of common factors. This turns a seemingly complicated expression into a more approachable form when combined using algebraic properties.

Understanding and working with radicals is essential in algebra and beyond. They are foundational to topics like quadratic equations and calculus. Therefore, becoming comfortable with radicals, and knowing when and how to simplify them using common factors, is a valuable skill in your mathematical toolkit.