Rationalize each denominator. See Example 8. 3———— 4 + √5

Verified step by step guidance

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

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, if the denominator contains a square root, you can multiply by the same square root to simplify the expression.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between its two terms. For instance, the conjugate of (a + b) is (a - b). When rationalizing denominators that contain square roots, multiplying by the conjugate can help eliminate the square root, leading to a simpler expression. This technique is particularly useful when the denominator is a sum or difference involving a square root.

Simplifying Radicals

Simplifying radicals involves reducing a square root or other root to its simplest form. This can include factoring out perfect squares from under the radical sign or rewriting the expression in a way that minimizes the radical's complexity. Understanding how to simplify radicals is essential for effectively rationalizing denominators and ensuring that the final expression is in its simplest form.