Problem 51 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ \sqrt{3 x}(\sqrt{3}-\sqrt{x}) $$

Step-by-Step Solution

Answer

The product is simplified to \( 3\sqrt{x} - x\sqrt{3} \).

Step 1: Distribute the Square Root

To multiply the expression, first distribute \( \sqrt{3x} \) across the terms inside the parentheses \( (\sqrt{3} - \sqrt{x}) \). This means you will multiply \( \sqrt{3x} \) by each term separately.

Step 2: Multiply \( \sqrt{3x} \) by \( \sqrt{3} \)

First, multiply \( \sqrt{3x} \) by \( \sqrt{3} \). The result is: \( \sqrt{3x} \times \sqrt{3} = \sqrt{3 \times 3x} = \sqrt{9x} \). Simplifying \( \sqrt{9x} \) gives \( 3\sqrt{x} \) since \( \sqrt{9} = 3 \).

Step 3: Multiply \( \sqrt{3x} \) by \( -\sqrt{x} \)

Now, multiply \( \sqrt{3x} \) by \( -\sqrt{x} \). The result is: \( \sqrt{3x} \times \sqrt{x} = \sqrt{3x^2} \). Simplifying this gives \( \sqrt{3}x \) because \( \sqrt{x^2} = x \).

Step 4: Combine the Terms

Combine the simplified terms from Step 2 and Step 3. The expression becomes: \( 3\sqrt{x} - x\sqrt{3} \).

Step 5: Final Simplification

No further simplification is possible because \( 3\sqrt{x} \) and \( x\sqrt{3} \) are unlike terms. Thus, the simplified expression is \( 3\sqrt{x} - x\sqrt{3} \).

Key Concepts

Understanding the Distributive Property

The distributive property is a key algebraic rule used to simplify expressions and equations. It allows you to multiply a single term by each term within a set of parentheses. In this exercise, the distributive property helps us multiply the term \( \sqrt{3x} \) by each component of the expression \( (\sqrt{3} - \sqrt{x}) \). This process involves following these steps:

• Multiplying \( \sqrt{3x} \) by \( \sqrt{3} \).
• Multiplying \( \sqrt{3x} \) by \( -\sqrt{x} \).

You then add these products together, resulting in an expression that is easier to simplify. This method not only organizes calculations but also aids in error reduction in longer expressions involving radicals.

Steps to Simplifying Expressions

Simplifying expressions involves combining like terms and reducing calculations to their simplest form. In this exercise, you start by using the distributive property to expand the expression. Each multiplication gives a new radical expression:

• \( \sqrt{9x} \) simplifies to \( 3\sqrt{x} \) because \( \sqrt{9} \) results in \( 3 \).
• \( \sqrt{3x^2} \) reduces to \( \sqrt{3}x \) because \( \sqrt{x^2} \) equals \( x \).

After getting the products, you proceed to combine them into a single expression \( 3\sqrt{x} - x\sqrt{3} \). Simplification ensures that the final expression retains only necessary operations and terms, making it as straightforward as possible.

Breaking Down Square Roots

Square roots are a fundamental concept in math, indicating a number which when multiplied by itself gives the original number. For instance, the square root of \( 9 \) is \( 3 \) because \( 3 \times 3 = 9 \). Similarly, within expressions, square roots require manipulation:

• \( \sqrt{9x} = 3\sqrt{x} \) since \( \sqrt{9} \) equals \( 3 \).
• \( \sqrt{3x^2} = \sqrt{3}x \) because \( \sqrt{x^2} = x \).

Understanding and simplifying square roots helps achieve a cleaner expression. It's crucial to simplify them correctly to effectively evaluate the radical expressions involved.

Working with Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations that must adhere to algebraic rules. In this exercise, multiplying radicals involves combining variables (like \( x \)) with square roots (like \( \sqrt{3} \)). Techniques such as the distributive property help manage these processes better.

• Apply rules to multiply similar terms.
• Combine simplified terms into a more manageable form.

Understanding algebraic guidelines enables you to work through complex expressions involving variables and radicals efficiently. This exercise illustrates how critical these techniques are in tackling problems involving multiplication of radicals.