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

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{6}{\sqrt{5}}\).
2. Recall that to rationalize a denominator containing a square root, multiply both numerator and denominator by the same square root to eliminate the radical in the denominator.
3. Multiply numerator and denominator by \(\sqrt{5}\): \(\frac{6}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\).
4. Simplify the numerator: \(6 \times \sqrt{5} = 6\sqrt{5}\).
5. Simplify the denominator: \(\sqrt{5} \times \sqrt{5} = 5\), so the expression becomes \(\frac{6\sqrt{5}}{5}\).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is done by multiplying both numerator and denominator by a suitable expression that will make the denominator a rational number, often the conjugate or the radical itself.

Properties of Square Roots

Square roots have properties such as √a × √a = a, which are used to simplify expressions. Understanding how to manipulate square roots allows you to convert irrational denominators into rational numbers by using these properties during multiplication.

Multiplying Fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. When rationalizing, you multiply the fraction by a form of 1 (like √5/√5) to keep the value unchanged while simplifying the denominator.