Problem 54 - Precalculus Mathematics for Calculus

Question

Perform the indicated operations and simplify. $$\left(x^{1 / 2}+y^{1 / 2}\right)\left(x^{1 / 2}-y^{1 / 2}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Recognize the Expression

The expression \((x^{1 / 2} + y^{1 / 2})(x^{1 / 2} - y^{1 / 2})\) is a difference of squares in the form \((a+b)(a-b)\), where \(a = x^{1/2}\) and \(b = y^{1/2}\).

Step 2: Apply the Difference of Squares Formula

The formula for the difference of squares is \((a+b)(a-b) = a^2 - b^2\). Plugging in \(a = x^{1/2}\) and \(b = y^{1/2}\), we get \((x^{1/2})^2 - (y^{1/2})^2\).

Step 3: Simplify the Expression

Simplify \((x^{1/2})^2 - (y^{1/2})^2\) to get \(x - y\). This is the result of squaring \(x^{1/2}\) and \(y^{1/2}\).

Key Concepts

Radical Expressions

Radical expressions might seem complex at first, but they are foundational in algebra. These expressions involve roots—most commonly square roots. When you see something like \(x^{1/2}\), it's simply another way of writing \(\sqrt{x}\). Similarly, \(y^{1/2}\) is \(\sqrt{y}\). Understanding this notation is crucial because it allows you to manipulate and simplify different expressions easily.

• Radical expressions are often found in physics and engineering problems.
• They might look complicated but can be simplified using basic algebraic rules.

When dealing with radical expressions, always ensure you're comfortable with these basic conversions first.
This way, you can recognize patterns and apply the right mathematical rules to simplify them.

Simplification

Simplification is a powerful tool in mathematics. It allows you to break down complex expressions into simpler, more manageable forms. In our current exercise, simplification was done by recognizing a pattern called the "difference of squares." The expression \((x^{1 / 2}+y^{1 / 2})(x^{1 / 2}-y^{1 / 2})\) is actually ideal for this simplification method.
When simplifying:

• Identify any patterns or common algebraic forms, such as the difference of squares.
• Use known formulas to simplify quickly, like \((a+b)(a-b) = a^2 - b^2\).

By applying simplification, leveraging algebraic principles, the expression reduces to the simpler form \(x - y\). This reduces unnecessary complexity and makes further algebraic manipulations more straightforward.
As you advance in algebra, simplification will save time and reduce the chance of mistakes.

Algebraic Expressions

Algebraic expressions are essentially mathematical phrases that contain numbers, variables, and operators. They are the backbone of algebra, found in almost every problem you tackle. Understanding how to manipulate and operate on these expressions is vital.
In the exercise provided, we dealt with an algebraic expression involving radicals:

• Initial understanding revolves around recognizing the components: terms like \(x^{1/2}\) and \(y^{1/2}\).
• It is crucial to understand how each term operates within the expression.

The solution showed us how to use the difference of squares, a particular algebraic identity, to simplify our given expression.
It highlights the importance of recognizing patterns and rules that can be applied to reduce complexity.
Mastery over algebraic expressions requires practice and a good understanding of foundational concepts like radicals and simplification techniques.