Problem 102 - Elementary and Intermediate Algebra

Question

Rationalize each numerator. All variables represent positive real numbers. $$ \frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}} $$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{x - y}{x + 2\sqrt{xy} + y} \).

Step 1: Understand the Problem

The problem asks us to rationalize the numerator of the expression \( \frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}} \). This means we need to eliminate the square root from the numerator by multiplying both the numerator and the denominator by a suitable expression.

Step 2: Identify the Conjugate

To rationalize the numerator \( \sqrt{x}-\sqrt{y} \), we use its conjugate, \( \sqrt{x}+\sqrt{y} \). Multiplying the numerator by its conjugate will help us simplify it.

Step 3: Multiply by the Conjugate

Multiply both the numerator and the denominator by \( \sqrt{x}+\sqrt{y} \), which is the conjugate of the numerator: \[\frac{(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})}{(\sqrt{x}+\sqrt{y})(\sqrt{x}+\sqrt{y})}\]

Step 4: Simplify the Numerator

The numerator simplifies using the difference of squares formula: \[(\sqrt{x})^2 - (\sqrt{y})^2 = x - y\]

Step 5: Simplify the Denominator

The denominator is \[(\sqrt{x}+\sqrt{y})^2 = x + 2\sqrt{xy} + y\]

Step 6: Write the Final Simplified Expression

The expression after rationalizing the numerator is: \[\frac{x - y}{x + 2\sqrt{xy} + y}\]

Key Concepts

Conjugate Multiplication

Conjugate multiplication is a powerful technique in algebra that helps us eliminate radicals from denominators or numerators. To understand it better, think of a conjugate as a modified version of an expression. If we have an expression like \(a + b\), its conjugate is \(a - b\). Notice that the sign between \(a\) and \(b\) changes. This change is crucial in the rationalization process.

When dealing with radicals, multiplying a numerator or denominator by its conjugate can simplify the expression. This is particularly useful when we want to remove square roots. For example, in the expression \(\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\), the conjugate of \(\sqrt{x}-\sqrt{y}\) is \(\sqrt{x}+\sqrt{y}\). By multiplying the entire fraction by \(1\) expressed as \(\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}+\sqrt{y}}\), the radicals in the numerator are eliminated. This changes the expression into something that can be worked with more easily.

Here’s a tip! Always remember to multiply both the numerator and the denominator by the conjugate; otherwise, the equality will not be maintained.

Difference of Squares Formula

The difference of squares is a handy formula used in various mathematical contexts, particularly in simplifying expressions containing a pair of squared terms separated by a minus sign. The formula is written as \(a^2 - b^2 = (a-b)(a+b)\). This formula exploits the principle that the product of conjugate pairs always results in a difference of squares.

In the context of our problem, using the formula helps simplify the expression \((\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})\). Here, we square both terms \(\sqrt{x}\) and \(\sqrt{y}\), and apply the formula: \((\sqrt{x})^2 - (\sqrt{y})^2 = x - y\). This result is a simpler form without any radicals.

The beauty of the difference of squares formula lies in its simplicity and power. It effectively converts tricky radical multiplication into straightforward difference calculations that are much simpler to manage.

Simplifying Radical Expressions

Simplifying radical expressions is akin to tidying up a cluttered room; it makes complex expressions much easier to read and work with. The goal of simplifying is to have no radicals in the numerator or to express them as simply as possible.

Let's break down the simplification process using the given example. Initially, we had: \(\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\). The inversion of radicals from the numerator was accomplished through conjugate multiplication.

• First, identify expressions that can be simplified using known algebraic identities, such as the difference of squares.
• Multiply the numerator by its conjugate to eliminate the radicals.
• Simplify the result to obtain \(x-y\) in the numerator and perform the same multiplication in the denominator to obtain \(x + 2\sqrt{xy} + y\).

Remember, radical expressions might look intimidating initially, but by using techniques such as conjugate multiplication and algebraic formulas, you can transform them into simpler, more manageable forms.