Problem 38 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(3m - 2n\).

Step 1: Recognize the Expression Format

The expression to be simplified, \((\sqrt{3m} + \sqrt{2n})(\sqrt{3m} - \sqrt{2n})\), is a difference of squares. Recall the identity: \((a + b)(a - b) = a^2 - b^2\). Our task is to apply this to the expression.

Step 2: Identify "a" and "b"

In our case, use the identity by recognizing \(a\) as \(\sqrt{3m}\) and \(b\) as \(\sqrt{2n}\). We will substitute these values into the formula: \(a^2 - b^2\).

Step 3: Calculate \(a^2\)

Calculate \(a^2\) by squaring \(\sqrt{3m}\): \(a^2 = (\sqrt{3m})^2 = 3m\).

Step 4: Calculate \(b^2\)

Calculate \(b^2\) by squaring \(\sqrt{2n}\): \(b^2 = (\sqrt{2n})^2 = 2n\).

Step 5: Apply the Difference of Squares Formula

Substitute \(a^2 = 3m\) and \(b^2 = 2n\) into \(a^2 - b^2\): \(3m - 2n\).

Step 6: Final Simplified Expression

The simplified expression of \((\sqrt{3m} + \sqrt{2n})(\sqrt{3m} - \sqrt{2n})\) is \(3m - 2n\).

Key Concepts

Difference of Squares

The difference of squares is a useful algebraic identity that allows us to simplify expressions of the form \((a + b)(a - b)\). This identity states that \((a + b)(a - b) = a^2 - b^2\). Essentially, when you multiply a sum by a difference of the same two terms, you end up with their squares' difference.

In our exercise, the expression \((\sqrt{3m} + \sqrt{2n})(\sqrt{3m} - \sqrt{2n})\) is set up perfectly for this identity. Here, \(a = \sqrt{3m}\) and \(b = \sqrt{2n}\). By using the identity, instead of performing lengthy multiplication, we can directly compute:

• SQUARE \(a\): \((\sqrt{3m})^2 = 3m\)
• SQUARE \(b\): \((\sqrt{2n})^2 = 2n\)

Thus, the expression simplifies to \(3m - 2n\). Recognizing and applying the difference of squares can greatly streamline calculations, especially when dealing with more complex algebraic expressions.

Radicals

Radicals, often represented as "\(\sqrt{\ldots}\)", denote the concept of extracting the root of a number or expression. The most common radical is the square root, denoted as \(\sqrt{\ldots}\). Radicals are handy for simplifying expressions involving roots and can sometimes be further simplified or manipulated depending on the context.

In our exercise, we encountered radicals in the terms \(\sqrt{3m}\) and \(\sqrt{2n}\). When you square a radical, such as \((\sqrt{x})^2\), you get the original number or expression \(x\). This property is paramount when working with differences of squares, as it helps simplify those expressions efficiently.

This section of mathematics empowers students to work flexibly with algebraic expressions that include roots, significantly broadening their problem-solving toolbox.

Multiplication of Binomials

Multiplication of binomials is the process of expanding two expressions, each consisting of two terms, and simplifying the resulting expression. Typically, for binomials of the form \((a + b)(c + d)\), we use the distributive property to expand:

• Multiply: \((a \cdot c), (a \cdot d), (b \cdot c), (b \cdot d)\)
• Combine like terms

In this specific exercise, we don't perform a straightforward multiplication because the formula for the difference of squares enables a shortcut. However, understanding multiplication of binomials is crucial because it forms the basis of recognizing patterns like the difference of squares.

Grasping how to distribute each term ensures a solid foundation in algebra, allowing students to navigate more complex expressions and identities with ease.