Problem 47 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((\sqrt{2 x}+\sqrt{3 y})(\sqrt{2 x}-\sqrt{3 y})\)

Step-by-Step Solution

Answer

The product is \(2x - 3y\).

Step 1: Identify the Expression Type

The given expression \((\sqrt{2x} + \sqrt{3y})(\sqrt{2x} - \sqrt{3y})\) fits the form of the difference of squares \((a + b)(a - b)\), where \(a = \sqrt{2x}\) and \(b = \sqrt{3y}\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \((a + b)(a - b) = a^2 - b^2\). Therefore, apply this formula to the given expression: \[(\sqrt{2x} + \sqrt{3y})(\sqrt{2x} - \sqrt{3y}) = (\sqrt{2x})^2 - (\sqrt{3y})^2\]

Step 3: Square the Terms

Compute the squares of \(\sqrt{2x}\) and \(\sqrt{3y}\): \((\sqrt{2x})^2 = 2x\) and \((\sqrt{3y})^2 = 3y\).

Step 4: Substitute and Simplify

Substitute the squared terms back into the expression:\((\sqrt{2x})^2 - (\sqrt{3y})^2 = 2x - 3y\).Thus, the simplest radical form of the product is \(2x - 3y\).

Key Concepts

Difference of Squares

When you come across a math expression like \((\sqrt{2 x}+\sqrt{3 y})(\sqrt{2 x}-\sqrt{3 y})\), what's happening might seem a little tricky. But it becomes much easier when you spot the difference of squares pattern. The difference of squares is a special algebraic formula: \((a + b)(a - b) = a^2 - b^2\). This formula helps you simplify expressions involving radicals by breaking them down into simpler parts.

• You must identify expressions that look like \((a + b)(a - b)\)
• Notice that the middle terms cancel out, leaving you with just the squares: \(a^2 - b^2\).

In the exercise we're looking at, you identify \(a = \sqrt{2x}\) and \(b = \sqrt{3y}\). Then it's just a matter of applying the formula to transform it into something easier to handle, which ends up being the straightforward expression \(2x - 3y\).

Radical Expressions

When dealing with expressions that include square roots or any roots, they are known as radical expressions. Radicals are often written using the radical symbol (√). In our example, \(\sqrt{2x}\) and \(\sqrt{3y}\) are both radical expressions.Handling these kinds of expressions involves a few key ideas:

• Understanding that the radical represents the solution to a number that can be multiplied by itself to give another number.
• The expression \(\sqrt{2x}\) means you are looking for the number which gives \(2x\) when squared.
• When working with radicals, always consider whether you can simplify them. In this particular exercise, they were already in their simplest form.

Learning how to manage and manipulate radicals is immensely helpful not just for solving homework but also for developing a deeper comprehension of algebraic concepts.

Expressing in Simplest Form

Expressing any mathematical expression in its simplest form makes it easier to work with. In mathematics, especially when dealing with radical expressions, simplicity leads to clarity. The end goal is to reduce the expression as much as possible while retaining its original value.So, what does expressing in simplest form mean?

• It involves performing operations such as multiplication, division, addition, or subtraction and then simplifying the expression by canceling out terms that don't need to be there.
• In the case of our expression \((\sqrt{2x} + \sqrt{3y})(\sqrt{2x} - \sqrt{3y})\), it is already simplified using the difference of squares to \(2x - 3y\).
• Simplifying expressions can involve finding common factors, canceling denominators, or reducing within square roots.

Being able to express things simply is an essential mathematical skill that can boost problem-solving efficiency and improve understanding.