Problem 72 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Goal

The goal is to rationalize the numerator of the expression \(\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}\). This means we need to eliminate the radicals in the numerator.

Step 2: Multiply by the Conjugate

To rationalize the numerator, you should multiply both the numerator and the denominator by the conjugate of the numerator. The conjugate of \(\sqrt{x}+\sqrt{y}\) is \(\sqrt{x}-\sqrt{y}\). Thus, multiply:\[\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}} \times \frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}-\sqrt{y}}\]

Step 3: Simplify the Numerator

Multiply the numerator:\[ (\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y}) = (\sqrt{x})^2 - (\sqrt{y})^2 = x - y \]This step uses the difference of squares formula: \((a+b)(a-b) = a^2-b^2\).

Step 4: Simplify the Denominator

Multiply the denominator:\[(\sqrt{x}-\sqrt{y})^2 = (\sqrt{x})^2 - 2\cdot\sqrt{x}\cdot\sqrt{y} + (\sqrt{y})^2 = x - 2\sqrt{xy} + y\]This expansion uses the identity \((a-b)^2 = a^2 - 2ab + b^2\).

Step 5: Write the Rationalized Expression

After simplifying both the numerator and the denominator, the rationalized expression is:\[\frac{x-y}{x-2\sqrt{xy}+y}\]

Step 6: Verify the Result

The numerator is free of radicals, which was the goal. The expression is now rationalized successfully.

Key Concepts

Difference of Squares

The difference of squares is a powerful algebraic tool that helps in the simplification of expressions, particularly when dealing with radicals or variables. The formula for the difference of squares is expressed as \((a+b)(a-b) = a^2 - b^2\).

This formula is especially useful because it allows us to transform a product of binomial conjugates into a simpler expression. In the example exercise, when we multiplied the numerator \((\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})\), the radicals disappeared, leaving us with \(x-y\). This simplification is a direct result of applying the difference of squares formula.

Key points to remember:

• The difference of squares works specifically with conjugate pairs.
• The result is always the difference of two perfect squares.

Conjugate

A conjugate in mathematics refers to a binomial formed by changing the sign between two terms. For example, the conjugate of \(a+b\) is \(a-b\).

In rationalization processes, using conjugates is vital because they help eliminate radicals from numerators or denominators. When dealing with square roots, multiplying by the conjugate effectively uses the difference of squares to cancel out radicals.

In our original step-by-step solution, we multiplied the expression \(\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}\) by the conjugate of the numerator, \(\sqrt{x}-\sqrt{y}\), which facilitated the removal of radicals from the numerator.

Key points on the use of conjugates:

• Conjugates apply to expressions involving two terms.
• They are instrumental in eliminating irrational components.
• Always involve flipping the sign between two terms.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest form. This could involve eliminating radicals, combining like terms, or using algebraic identities.

In many algebra problems, such as rationalizing, simplifying ensures that expressions are more manageable and easier to understand. During our exercise, we saw simplification in action when applying both the difference of squares and conjugates.

It's crucial that each step in the simplification follows mathematical rules. For instance, multiplying by the conjugate in our problem was pivotal in simplifying the complex algebraic fraction we began with.

Here's what to focus on when simplifying expressions:

• Identify and apply algebraic identities.
• Combine like terms where possible.
• Follow a systematic approach to ensure accuracy.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both consist of algebraic expressions rather than just numbers. Simplifying algebraic fractions often involves complex algebraic identities and techniques like factoring or rationalization.

In the given exercise, the expression \(\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}\) is an algebraic fraction with square roots in the numerator and denominator.

The goal was to rationalize, or remove, the radical from the numerator by using conjugates. This involves finding an equivalent fraction where the numerator is free of radicals, thus greatly simplifying computations.

Important aspects of algebraic fractions include:

• Understanding how to handle complex expressions consisting of variables and radicals.
• Knowing techniques such as factoring, conjugates, and identities for simplification.
• Recognizing that simplifying these fractions expands mathematical understanding and versatility.