In Exercises 93–104, rationalize each numerator. Simplify, if possible.√a - √b √a + √b

Verified step by step guidance

1. Identify the expression to rationalize: \( \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}} \).
2. To rationalize the numerator, multiply both the numerator and the denominator by the conjugate of the numerator: \( \sqrt{a} + \sqrt{b} \).
3. The expression becomes: \( \frac{(\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})}{(\sqrt{a} + \sqrt{b})(\sqrt{a} + \sqrt{b})} \).
4. Apply the difference of squares formula to the numerator: \((\sqrt{a})^2 - (\sqrt{b})^2 = a - b\).
5. Simplify the denominator: \((\sqrt{a} + \sqrt{b})^2 = a + 2\sqrt{a}\sqrt{b} + b\).

Key Concepts

Rationalizing the Numerator

Rationalizing the numerator involves eliminating any square roots or irrational numbers from the numerator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by the conjugate of the numerator. In this case, the conjugate of (√a - √b) is (√a + √b), which helps in simplifying the expression while maintaining its value.

Conjugates

Conjugates are pairs of binomials that differ only in the sign between their terms. For example, the conjugate of (√a - √b) is (√a + √b). When multiplied together, conjugates yield a difference of squares, which simplifies the expression and eliminates the square roots in the numerator, making it easier to work with.

Simplifying Radicals

Simplifying radicals involves reducing square roots to their simplest form, which can include factoring out perfect squares. This process makes expressions more manageable and easier to interpret. In the context of rationalizing, simplifying the resulting expression after multiplication can lead to a clearer final answer.