Problem 111 - College Algebra with Modeling and Visualization

Question

Rationalize the denominator. $$ \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}} $$

Step-by-Step Solution

Answer

\( \frac{a + 2\sqrt{ab} + b}{a - b} \) is the rationalized expression.

Step 1: Identify the Conjugate

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( \sqrt{a} - \sqrt{b} \) is \( \sqrt{a} + \sqrt{b} \).

Step 2: Multiply by the Conjugate

Multiply the numerator and the denominator of the expression by the conjugate \( \sqrt{a} + \sqrt{b} \). So, the expression becomes:\[\frac{(\sqrt{a} + \sqrt{b})(\sqrt{a} + \sqrt{b})}{(\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})}\].

Step 3: Simplify the Denominator

Use the difference of squares formula: \((\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b}) = a - b\).Thus, the denominator simplifies to \(a - b\).

Step 4: Expand the Numerator

Expand the numerator by using the distributive property:\((\sqrt{a} + \sqrt{b})(\sqrt{a} + \sqrt{b}) = (\sqrt{a})^2 + 2\sqrt{a}\sqrt{b} + (\sqrt{b})^2 = a + 2\sqrt{ab} + b\).

Step 5: Write the Final Expression

Now, write down the rationalized expression as:\[\frac{a + 2\sqrt{ab} + b}{a - b}\].

Key Concepts

Conjugate

The concept of a conjugate is vital when rationalizing denominators in expressions with square roots. The conjugate of a binomial expression like \( \sqrt{a} - \sqrt{b} \) involves changing the sign between the two terms. Therefore, the conjugate here is \( \sqrt{a} + \sqrt{b} \). Using conjugates, we aim to eliminate the square root from the denominator.

By multiplying the numerator and the denominator by the conjugate, we make use of special product identities, turning the denominator into a form that no longer contains a radical. This makes the expression much easier to handle, especially in further calculations or evaluations. Think of the conjugate as a tool that balances the expression without changing its value, because multiplying by a form of 1 (like \( \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}} \)) does not affect the overall expression.

Difference of Squares

The difference of squares is a specific formula used commonly in algebra: \[(x + y)(x - y) = x^2 - y^2.\]This formula simplifies the product of two binomials that are conjugates (such as \( \sqrt{a} + \sqrt{b} \) and \( \sqrt{a} - \sqrt{b} \)).

Applying this formula to \((\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})\), we get:

• \( (\sqrt{a})^2 - (\sqrt{b})^2 = a - b \)

This process of cancelling out the square root results in just the difference \(a - b\) in the denominator.

The beauty of the difference of squares is in its simplicity, allowing complex expressions with radicals to become straightforward and simplified.

Simplifying Expressions

Simplifying expressions involves several algebraic techniques to rewrite an expression in a reduced form. One common aspect is to eliminate radicals from denominators by rationalizing them. This is done by using conjugates and the difference of squares as seen in the earlier steps.

In the example \( \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}} \), after multiplying by the conjugate and using the difference of squares, the expression becomes:

• Numerator: \( (\sqrt{a} + \sqrt{b})^2 = a + 2\sqrt{ab} + b \)
• Denominator: \( a - b \)

So, the final simplified form is:\[\frac{a + 2\sqrt{ab} + b}{a - b}.\]

Simplifying makes expressions more manageable and often prepares them for further use in equations or functions. It's important to remember each step retains the value of the expression, altering only its appearance for easy understanding and use.