Rationalize each denominator. See Example 8. 4—— √6

Verified step by step guidance

1. Identify the expression to rationalize: \( \frac{4}{\sqrt{6}} \).
2. Multiply both the numerator and the denominator by \( \sqrt{6} \) to eliminate the square root in the denominator.
3. This results in: \( \frac{4 \cdot \sqrt{6}}{\sqrt{6} \cdot \sqrt{6}} \).
4. Simplify the denominator: \( \sqrt{6} \cdot \sqrt{6} = 6 \).
5. The expression becomes: \( \frac{4\sqrt{6}}{6} \). Simplify further if possible.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers from the denominator of a fraction. This is typically done by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, to rationalize a fraction with a square root in the denominator, you multiply by the same square root.

Properties of Square Roots

Understanding the properties of square roots is essential for rationalizing denominators. The key property is that the square root of a product can be expressed as the product of the square roots, i.e., √(a*b) = √a * √b. This property allows us to manipulate expressions involving square roots effectively, facilitating the rationalization process.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This is important after rationalizing the denominator, as it ensures the final expression is as concise as possible. Simplification often involves factoring and canceling common terms in the numerator and denominator.