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

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{5}{\sqrt{5}}\).
2. Recall that rationalizing the denominator means eliminating the square root from the denominator by multiplying numerator and denominator by the same radical.
3. Multiply both numerator and denominator by \(\sqrt{5}\) to get: \(\frac{5}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\).
4. Use the property \(\sqrt{a} \times \sqrt{a} = a\) to simplify the denominator: \(\sqrt{5} \times \sqrt{5} = 5\).
5. Rewrite the expression as \(\frac{5 \times \sqrt{5}}{5}\) and then simplify by canceling the common factor in numerator and denominator.

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 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.