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

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Identify the expression to rationalize: \(\frac{6}{\sqrt{5}}\).

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.

Multiply numerator and denominator by \(\sqrt{5}\): \(\frac{6}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\).

Simplify the numerator: \(6 \times \sqrt{5} = 6\sqrt{5}\).

Simplify the denominator: \(\sqrt{5} \times \sqrt{5} = 5\), so the expression becomes \(\frac{6\sqrt{5}}{5}\).

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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.

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Rationalizing Denominators

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.

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Imaginary Roots with the Square Root Property

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.

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