Problem 51 - Intermediate Algebra

Question

Rationalize each denominator. See Example 4. $$ \frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}} $$

Step-by-Step Solution

Answer

\( \frac{x - \sqrt{xy}}{x - y} \)

Step 1: Identify the Conjugate

The expression given is \( \frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}} \). To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator, which is \( \sqrt{x}-\sqrt{y} \).

Step 2: Multiply by the Conjugate

Multiply the expression by \( \frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}-\sqrt{y}} \):\[\frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}} \cdot \frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}-\sqrt{y}}\]

Step 3: Simplify the Denominator

The denominator becomes a difference of squares:\[(\sqrt{x} +\sqrt{y})(\sqrt{x} - \sqrt{y}) = x - y\]

Step 4: Simplify the Numerator

The numerator is expanded as follows:\[\sqrt{x}(\sqrt{x} - \sqrt{y}) = x - \sqrt{xy}\]

Step 5: Write the Rationalized Expression

Combine the simplified numerator and denominator:\[\frac{x - \sqrt{xy}}{x - y}\]

Step 6: Confirm Simplification

The expression \( \frac{x - \sqrt{xy}}{x - y} \) is now fully simplified with a rationalized denominator.

Key Concepts

Conjugate Pairs

When dealing with irrational denominators, like those containing square roots, a common strategy to simplify them is utilizing the conjugate pair. The conjugate of an expression \(a + b\) is \(a - b\), and vice-versa. This technique is specifically useful because it leverages the difference of squares identity, which eliminates the square roots in the denominator.

• For example, the conjugate of \(\sqrt{x} + \sqrt{y}\) is \(\sqrt{x} - \sqrt{y}\).
• Multiplying by this conjugate will help us remove the square roots from the denominators.

By multiplying the original fraction by the conjugate of its denominator (appropriately set as a fraction equivalent to 1), we achieve a new denominator that is a rational number, making further calculations simpler.

Difference of Squares

The difference of squares is a fundamental algebraic identity often used when rationalizing denominators with conjugate pairs. The identity can be written as:\[ (a+b)(a-b) = a^2 - b^2 \]

• In the case of \(\sqrt{x} + \sqrt{y}\) and \(\sqrt{x} - \sqrt{y}\), applying the difference of squares results in \(x - y\), a simplified integer or rational number.
• This mechanism nullifies the square roots, effectively transforming them into whole numbers.

Through this conversion, we not only simplify fractions but also make the expressions easier to handle.

Simplifying Expressions

Simplifying expressions is a crucial part of working with algebraic fractions and involves reducing expressions to their simplest form while preserving their worth.

• Upon rationalizing a denominator using conjugate pairs and the difference of squares, it's essential to simplify the entire expression for the most clarified result.
• After applying the conjugate, the expression in the example: \(\frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}}\) becomes \(\frac{x - \sqrt{xy}}{x-y}\).

The simplification process focuses on expressing the numerator and denominator in a form that is generally easier to interpret and solve, ensuring that the expression reflects its most reduced state, void of any radicals in the denominator.