Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. 1/(2 + √5)

Verified step by step guidance

1. Identify the expression to rationalize: \( \frac{1}{2 + \sqrt{5}} \).
2. Multiply the numerator and the denominator by the conjugate of the denominator: \( 2 - \sqrt{5} \).
3. Write the expression as: \( \frac{1}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}} \).
4. Simplify the denominator using the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\), where \(a = 2\) and \(b = \sqrt{5}\).
5. Simplify the numerator and the denominator separately to complete the rationalization process.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers from the denominator of a fraction. This is typically achieved 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 denominator like (2 + √5), one would multiply by the conjugate (2 - √5).

Conjugates

The conjugate of a binomial expression is formed by changing the sign of the second term. For instance, the conjugate of (a + b) is (a - b). In the context of rationalizing denominators, using the conjugate helps to simplify expressions by utilizing the difference of squares, which eliminates the square root when multiplied together.

Properties of Square Roots

Understanding the properties of square roots is essential for simplifying expressions involving them. Specifically, the property that √a * √a = a allows for the simplification of terms when rationalizing denominators. This property is crucial when multiplying by the conjugate, as it helps to transform the denominator into a rational number.