Problem 68 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (5 \sqrt{3 x}-\sqrt{y})(4 \sqrt{x}+1) $$

Step-by-Step Solution

Answer

The simplified form is \(20x \sqrt{3} + 5 \sqrt{3x} - 4 \sqrt{xy} - \sqrt{y}\).

Step 1: Expand the Expression

Use the distributive property to expand the expression: \( (5 \sqrt{3x} - \sqrt{y})(4 \sqrt{x} + 1) \)This gives:\[ 5 \sqrt{3x} \cdot 4 \sqrt{x} + 5 \sqrt{3x} \cdot 1 - \sqrt{y} \cdot 4 \sqrt{x} - \sqrt{y} \cdot 1 \]

Step 2: Multiply each term

Multiply the terms as follows:1. \( 5 \sqrt{3x} \cdot 4 \sqrt{x} = 5 \cdot 4 \cdot \sqrt{3x \cdot x} = 20 \cdot \sqrt{3x^2} \)2. \( 5 \sqrt{3x} \cdot 1 = 5 \sqrt{3x} \)3. \( -\sqrt{y} \cdot 4 \sqrt{x} = -4 \sqrt{y \cdot x} = -4 \sqrt{xy} \)4. \( -\sqrt{y} \cdot 1 = -\sqrt{y} \)

Step 3: Simplify using properties of square roots

Simplify each term:1. \( \sqrt{3x^2} = \sqrt{3} \cdot \sqrt{x^2} = \sqrt{3} \cdot x \), so \( 20 \sqrt{3x^2} = 20 \sqrt{3} \cdot x \)Rewriting the expanded expression:\[ 20x \sqrt{3} + 5 \sqrt{3x} - 4 \sqrt{xy} - \sqrt{y} \]

Step 4: Combine the terms

Combine like terms, which in this case involves rewriting to maintain order, as there are no like terms. The expression simplified is:\[ 20x \sqrt{3} + 5 \sqrt{3x} - 4 \sqrt{xy} - \sqrt{y} \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to break down complex expressions into simpler parts. It relates to how multiplication interacts with addition or subtraction. Imagine having a sum that you need to multiply by a number. Instead of first adding the values together, you can distribute the multiplication across each part of the sum.

In mathematical terms, the distributive property is represented as

• \( a(b + c) = ab + ac \)

This property ensures that each term of the sum inside the parentheses is multiplied by the term outside.

In the given problem,

• \((5 \sqrt{3x} - \sqrt{y})(4 \sqrt{x} + 1)\)

we apply the distributive property by multiplying every term in the first parenthesis with every term in the second parenthesis. This helps to systematically expand the entire expression step by step, making it easier to handle the complexity of the operations.

Square Roots

Square roots are symbols used to denote the operation that finds a value that, when multiplied by itself, results in the original number. The square root of a number \( a \) is written as \( \sqrt{a} \). For real numbers, every positive number has two square roots - one positive and one negative. However, in many algebraic expressions or real-world contexts, we only consider the positive root, which is why we only focus on \( \sqrt{a} \).

In our exercise, square roots appear in expressions like \( \sqrt{3x} \), \( \sqrt{x} \), and \( \sqrt{y} \). To simplify these expressions, we sometimes use properties of square roots, such as

• \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)

This allows us to break down complex square root terms into simpler parts.

For instance, \( 20 \sqrt{3x^2} \) is simplified using \( \sqrt{x^2} = x \) since \( x \times x = x^2 \). Thus, \( 20 \sqrt{3x^2} \) becomes \( 20 \cdot x \cdot \sqrt{3} \). Understanding these principles is crucial for simplifying algebraic expressions involving square roots by making them more manageable.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their most basic form while preserving their value. This process involves combining like terms, performing arithmetic operations, and applying algebraic rules such as the distributive property and properties of square roots.

To simplify an expression, you should:

• Combine like terms when possible, which are terms that have the same variables raised to the same power.
• Look for opportunities to factor or reduce operations using mathematical identities or properties.
• Simplify any complex fractions or irrational numbers using rational approximations or other mathematical techniques.

For the given exercise, once the expression is expanded, we rewrite it as

• \( 20x \sqrt{3} + 5 \sqrt{3x} - 4 \sqrt{xy} - \sqrt{y} \).

This is already simplified since there are no like terms to combine further.

Ultimately, knowing how to effectively simplify expressions will allow you to solve problems more quickly and understand the underlying relationships between the variables involved.