Problem 32 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{2 \sqrt{x}-5 \sqrt{y}}{\sqrt{x}+\sqrt{y}}\)

Step-by-Step Solution

Answer

\(\frac{2x + 5y - 7 \sqrt{xy}}{x - y}\) is the simplest form.

Step 1: Identify the Conjugate

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{x} + \sqrt{y}\) is \(\sqrt{x} - \sqrt{y}\).

Step 2: Multiply by the Conjugate

Multiply the fraction by \(\frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} - \sqrt{y}}\):\[\frac{(2 \sqrt{x} - 5 \sqrt{y})(\sqrt{x} - \sqrt{y})}{(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})}\]

Step 3: Simplify the Denominator

The denominator becomes a difference of squares:\[(\sqrt{x})^2 - (\sqrt{y})^2 = x - y\]

Step 4: Expand the Numerator

Use the distributive property to expand:\[(2 \sqrt{x} - 5 \sqrt{y})(\sqrt{x} - \sqrt{y}) = 2 \sqrt{x} \cdot \sqrt{x} - 2 \sqrt{x} \cdot \sqrt{y} - 5 \sqrt{y} \cdot \sqrt{x} + 5 \sqrt{y} \cdot \sqrt{y}\]

Step 5: Combine Like Terms in the Numerator

Simplify the terms:\[2x - 2 \sqrt{xy} - 5 \sqrt{xy} + 5y = 2x + 5y - 7 \sqrt{xy}\]

Step 6: Form the Final Fraction

Simplify the entire expression: \[\frac{2x + 5y - 7 \sqrt{xy}}{x - y}\]

Step 7: Simplify the Fraction if Possible

Check if any further simplification is possible. Since there are no common factors or additional simplifications, this is the simplest form.

Key Concepts

Conjugates in Rationalizing the Denominator

When you encounter a fraction with a radical in the denominator, the goal is to eliminate the radical. This process is known as "rationalizing the denominator." One of the most effective tools for this purpose is the use of conjugates.

The conjugate of a binomial expression like \(\sqrt{x} + \sqrt{y}\) is a critical concept. It involves changing the sign between the two terms, resulting in an expression \(\sqrt{x} - \sqrt{y}\). By multiplying the numerator and the denominator by this conjugate, you help remove radicals from the denominator.

Remember:

• The conjugate of \(a + b\) is \(a - b\).
• It is used to apply the difference of squares, which helps cancel out radicals.

You'll see this technique in action repeatedly when rationalizing complex expressions.

Difference of Squares Simplification

The difference of squares is a simple yet powerful algebraic identity, \(a^2 - b^2 = (a-b)(a+b)\). This identity is particularly useful when dealing with conjugates. When you multiply \(\sqrt{x} + \sqrt{y}\) by its conjugate \(\sqrt{x} - \sqrt{y}\), you create a difference of squares in the denominator.

Using the formula:

• \((\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y}) = x - y\)
• The radicals cancel out, leaving a simple expression.

Notice how this method cleanly removes the radicals, simplifying the fraction and making further algebraic manipulation straightforward. It's a key step, unlocking more efficient simplification paths.

The Distributive Property in Rational Expression Multiplication

The distributive property is essential in algebraic manipulation, particularly when expanding expressions. It states that \(a(b + c) = ab + ac\). When encountered with an expanded expression like \( (2 \sqrt{x} - 5 \sqrt{y})(\sqrt{x} - \sqrt{y})\), you apply this rule to multiply each term by every other term in the opposing binomial.

Here’s the expansion process:

• Multiply \(2 \sqrt{x}\) by both \(\sqrt{x}\) and \(\sqrt{y}\)
• Multiply \(-5 \sqrt{y}\) by both \(\sqrt{x}\) and \(\sqrt{y}\)

Using the distributive property ensures you combine all possible products, essential when forming the final expression. Simplifying these results leads you to \(2x + 5y - 7 \sqrt{xy}\), which is crucial to reaching the simplest form.

Simplifying Radicals Through Algebraic Manipulation

Simplifying radicals in expressions not only involves managing radicals but also systematically reducing complexity. Look at the expanded numerator: \(2 \sqrt{x}\cdot \sqrt{x}, -2 \sqrt{x}\cdot \sqrt{y}, -5 \sqrt{y}\cdot \sqrt{x}\), and \(5 \sqrt{y}\cdot \sqrt{y}\).

Simplification Steps include:

• Combine \(\sqrt{x} \cdot \sqrt{x} = x\) and \(\sqrt{y} \cdot \sqrt{y} = y\).
• Identify common like terms such as \(- 2 \sqrt{xy}\) and \(- 5 \sqrt{xy}\).
• Sum and simplify the terms to obtain \( 2x + 5y - 7 \sqrt{xy}\).

These steps reduce the numerator to a more workable form. This simplification is vital, fulfilling the purpose of rationalizing the denominator and attaining the expression's simplest version. This strategy not only eases the calculation process but also sharpens your algebra skills.