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

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

Recall that rationalizing the denominator means eliminating the square root from the denominator by multiplying numerator and denominator by the same radical.

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

Use the property \(\sqrt{a} \times \sqrt{a} = a\) to simplify the denominator: \(\sqrt{5} \times \sqrt{5} = 5\).

Rewrite the expression as \(\frac{5 \times \sqrt{5}}{5}\) and then simplify by canceling the common factor in numerator and denominator.

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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 by 1

Multiplying a fraction by a form of 1, such as √5/√5, changes the expression without altering its value. This technique is essential in rationalizing denominators because it allows you to introduce terms that simplify the denominator while keeping the fraction equivalent.

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