Problem 45 - Intermediate Algebra

Question

Rationalize each denominator. See Example 4. $$ \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}} $$

Step-by-Step Solution

Answer

The rationalized form is \(-5 + 2\sqrt{6}\).

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 \( \sqrt{2} + \sqrt{3} \) is \( \sqrt{2} - \sqrt{3} \).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \( \sqrt{2} - \sqrt{3} \). This gives us: \[ \frac{(\sqrt{2} - \sqrt{3})(\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})} \]

Step 3: Simplify the Numerator

Expand the numerator \((\sqrt{2} - \sqrt{3})(\sqrt{2} - \sqrt{3}) = (\sqrt{2})^2 - 2\sqrt{2}\sqrt{3} + (\sqrt{3})^2 = 2 - 2\sqrt{6} + 3 = 5 - 2\sqrt{6}.\)

Step 4: Simplify the Denominator

Expand the denominator using the difference of squares: \((\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3}) = (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1.\)

Step 5: Combine and Simplify

Combine the simplified numerator and denominator: \( \frac{5 - 2\sqrt{6}}{-1} = -5 + 2\sqrt{6}.\) This is the rationalized form of the expression.

Key Concepts

Understanding Conjugates

In mathematics, a **conjugate** is a simple yet powerful tool, especially for dealing with radical expressions or complex numbers. When you have a binomial expression such as \(a + b\), its conjugate is \(a - b\). Essentially, this involves changing the sign of the second term. Conjugates are pivotal for rationalizing denominators, making expressions simpler and easier to handle.Why do we focus on conjugates when rationalizing denominators? Here's how:

• Conjugates help eliminate radicals from the denominator. This is crucial because expressions are often more manageable when radicals are not in the denominator.
• By multiplying by the conjugate, you leverage the **difference of squares** formula, which tidily eliminates both middle terms and results in rational numbers.

For example, to rationalize the expression \(\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}}\), multiply both the numerator and denominator by the conjugate of the denominator \(\sqrt{2} - \sqrt{3}\). This transformation facilitates both simplification and subsequent calculations.

Deciphering Radical Expressions

Radical expressions include terms involving roots, such as square roots, cube roots, etc. These expressions can often look complicated but can be significantly simplified through some straightforward steps.Consider the radical expression \(\sqrt{2}+\sqrt{3}\). To break down these expressions:

• Identify the type of root involved—like a square root or cube root—that determines the fundamental behavior of the expression.
• When simplifying these, it can often involve operations such as multiplication, division, or finding conjugates.

In our example, we didn't just stick with a raw radical expression. We multiplied by its conjugate, a process that not only helped in simplifying the denominator but also turned the expression into a more approachable form. Mastery of radical expressions lies in understanding these manipulations, allowing you to rationalize denominators effectively.

The Power of Difference of Squares

The **difference of squares** is a crucial algebraic identity: \[(a + b)(a - b) = a^2 - b^2.\]This identity allows for easy simplification of expressions, particularly when working with conjugates. When we multiply a binomial expression by its conjugate, the middle terms cancel, simplifying the denominator dramatically when rationalizing. For instance, multiplying \(\sqrt{2} + \sqrt{3}\) by its conjugate \(\sqrt{2} - \sqrt{3}\) results in:

• The expression follows the difference of squares formula: \((\sqrt{2})^2 - (\sqrt{3})^2\).
• Calculating this results in \(2 - 3 = -1\).

Therefore, using the difference of squares can transform the denominator into a simple, rational number. This approach underscores the beauty of math in simplifying seemingly complex expressions while maintaining accuracy and elegance.