In Exercises 75–92, rationalize each denominator. Simplify, if possible. √b----------√a - √b

Verified step by step guidance

1. Step 1: Identify the expression that needs rationalization, which is \( \frac{\sqrt{b}}{\sqrt{a} - \sqrt{b}} \).
2. Step 2: To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{a} + \sqrt{b} \).
3. Step 3: Write the expression as \( \frac{\sqrt{b} \cdot (\sqrt{a} + \sqrt{b})}{(\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})} \).
4. Step 4: Simplify the denominator using the difference of squares formula: \((\sqrt{a})^2 - (\sqrt{b})^2 = a - b\).
5. Step 5: Expand the numerator: \( \sqrt{b} \cdot \sqrt{a} + \sqrt{b} \cdot \sqrt{b} = \sqrt{ab} + b \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, if the denominator is in the form of a square root, multiplying by the conjugate can help achieve this.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between its two terms. For instance, the conjugate of (√a - √b) is (√a + √b). Using the conjugate in rationalizing the denominator is effective because it utilizes the difference of squares formula, which simplifies the expression and eliminates the square roots in the denominator.

Simplifying Radicals

Simplifying radicals involves reducing a square root expression to its simplest form. This can include factoring out perfect squares from under the radical sign or combining like terms. Understanding how to simplify radicals is essential for effectively rationalizing denominators and ensuring that the final expression is as simplified as possible.