Rationalize each denominator. See Example 8. 6———— √5 + √3

Verified step by step guidance

1. Identify the expression to rationalize: \( \frac{6}{\sqrt{5} + \sqrt{3}} \).
2. Multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{5} - \sqrt{3} \).
3. Write the expression as: \( \frac{6(\sqrt{5} - \sqrt{3})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})} \).
4. Simplify the denominator using the difference of squares formula: \((\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2\).
5. Distribute the 6 in the numerator: \( 6\sqrt{5} - 6\sqrt{3} \), and simplify the expression to \( \frac{6\sqrt{5} - 6\sqrt{3}}{2} \).

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 a sum of square roots, one can multiply by the conjugate of that expression.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between the two terms. For instance, the conjugate of (a + b) is (a - b). When multiplying a binomial by its conjugate, the result is a difference of squares, which simplifies to a rational number. This technique is essential in rationalizing denominators that contain sums or differences of square roots.

Properties of Square Roots

Understanding the properties of square roots is crucial for simplifying expressions involving them. The key property states that √a * √b = √(a*b). This property allows for the combination of square roots and is often used when rationalizing denominators. Additionally, knowing that √(a + b) cannot be simplified directly into separate square roots is important for correctly manipulating expressions.