Problem 37 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ (\sqrt{3 x}-\sqrt{2 y})(\sqrt{3 x}+\sqrt{2 y}) $$

Step-by-Step Solution

Answer

The simplified expression is \(3x - 2y\).

Step 1: Recognize the Formula

The expression \((\sqrt{3x} - \sqrt{2y})(\sqrt{3x} + \sqrt{2y})\) is of the form \((a-b)(a+b)\), which is a difference of squares formula. This can be rewritten as \(a^2 - b^2\).

Step 2: Identify \(a\) and \(b\)

In this problem, \(a = \sqrt{3x}\) and \(b = \sqrt{2y}\). We need to square each of these terms.

Step 3: Apply the Difference of Squares

Using the formula \(a^2 - b^2\), calculate \(a^2 = (\sqrt{3x})^2\) and \(b^2 = (\sqrt{2y})^2\).

Step 4: Simplify Each Square

Calculate \(a^2 = (\sqrt{3x})^2 = 3x\) and \(b^2 = (\sqrt{2y})^2 = 2y\).

Step 5: Compute the Final Expression

Substitute the squared values back into the difference of squares: \(a^2 - b^2 = 3x - 2y\). This is the simplified form of the given expression.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions form the building blocks of algebra, allowing us to model and solve real-world problems through the use of symbols. In the exercise you encountered, the expression \[ (\sqrt{3x} - \sqrt{2y})(\sqrt{3x} + \sqrt{2y}) \]represents an algebraic expression.

Let's break it down:

• The expression consists of variables and constants grouped together using operations like addition, subtraction, or multiplication.
• The terms \(\sqrt{3x}\) and \(\sqrt{2y}\) involve both coefficients (3 and 2) and variables (\(x\) and \(y\)).
• Since the variables \(x\) and \(y\) are under a square root, they signify that these values are manipulated in a way that flips or scales their real number representation.

Each part of the expression plays a vital role in determining the overall value, making mastery of algebraic expressions crucial for math success.

Simplification Made Easy

Simplification is the process of reducing an expression to its most basic form without changing its value. In the exercise, you used the difference of squares formula to simplify.

Here's how simplification works in this context:

• Identify the pattern: The expression matches the difference of squares form \((a-b)(a+b)\), simplifying to \(a^2 - b^2\).
• Recognize each component: Determine which parts of the expression correspond to \(a\) and \(b\). In our case, \(a = \sqrt{3x}\) and \(b = \sqrt{2y}\).
• Apply mathematical rules: Calculate the squares of \(a\) and \(b\). This converts the expression into a simpler form, or \(3x - 2y\).

Simplification helps clarify the solution process, allowing complex operations to be more approachable and increasing ease of calculation.

The Role of Positive Real Numbers

In algebra, especially when dealing with expressions incorporating square roots, the domain of positive real numbers is fundamental.

Here's why positive real numbers are necessary:

• Consistency: In simplifying square roots, assuming that variables represent positive real numbers prevents negative results from causing undefined or complex expressions.
• Real-world Application: Often variables signify measurable quantities like distance or time, inherently non-negative, aligning with real numbers.
• Mathematical simplicity: Positive real numbers make calculations straightforward, avoiding complications from negatives during simplification.

When tackling exercises, knowing all variables are positive real numbers not only influences the approach to the problem but also reassures the applicability of mathematical techniques like simplification and the difference of squares.