Problem 38 - Elementary and 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 Pattern

Notice that the expression \((\sqrt{3m} + \sqrt{2n})(\sqrt{3m} - \sqrt{2n})\) follows the pattern \((a + b)(a - b) = a^2 - b^2\). Here, \(a = \sqrt{3m}\) and \(b = \sqrt{2n}\).

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula \((a + b)(a - b) = a^2 - b^2\), we can write: \((\sqrt{3m})^2 - (\sqrt{2n})^2\).

Step 3: Square Each Term

Calculate \((\sqrt{3m})^2 = 3m\) and \((\sqrt{2n})^2 = 2n\).

Step 4: Subtract the Squared Terms

Subtract the squared terms to simplify the expression: \(3m - 2n\).

Key Concepts

Difference of Squares

One of the most important patterns in algebra is the **difference of squares**. This is a method used to simplify expressions easily, especially those in the form

• a difference between two squares: \((a + b)(a - b) = a^2 - b^2\).
• In the expression provided, \(a = \sqrt{3m}\) and \(b = \sqrt{2n}\). By applying the difference of squares formula, we can multiply the terms inside the brackets without expanding everything step by step.

In our specific case, it changes a multiplication problem into a much simpler subtraction problem \((3m - 2n)\). This showcases how efficient this formula is for algebraic simplification.

Radicals

Radicals, often expressed with the square root sign (\(\sqrt{}\)), are an integral part of algebra. They represent the power of one-half. For instance:

• The square root of a number \(x\) is \(\sqrt{x}\), which can also be written as \(x^{1/2}\).
• In the given exercise, both \(\sqrt{3m}\) and \(\sqrt{2n}\) are radicals that represent numbers that when squared return the values inside the radical signs.

Working with radicals often involves simplification steps such as squaring these terms, thus eliminating the square root. This is why \((\sqrt{3m})^2 = 3m\) eliminates the radical, transforming it into an easier expression to work with in subtraction or addition contexts.

Simplification

Simplification in algebra is the process of transforming complex expressions into their simplest form. This makes equations and expressions easier to work with.

• By identifying patterns like the difference of squares, we can quickly reduce the complexity of expressions.
• In our task, using the difference of squares and understanding radicals helped us transform \((\sqrt{3m} + \sqrt{2n})(\sqrt{3m} - \sqrt{2n})\) into the simple expression \(3m - 2n\).

This shows that simplification doesn't just make expressions shorter; it paves the way for easier problem-solving, especially in more complex equations.

Positive Real Numbers

Positive real numbers are fundamental in ensuring that all given expressions remain valid, especially under operations like square roots. When we work with radicals:

• It's crucial to assume that the values inside the radicals represent positive real numbers. This is important because it guarantees that squaring a square root results in a positive value, like in \((\sqrt{3m})^2 = 3m\).
• In higher mathematics, considering positive real numbers also avoids complex numbers, greatly simplifying many algebraic processes for the student.

In algebraic expressions, these assumptions help ensure we avoid complications in computation and verify that expressions are consistently real and meaningful.