In Exercises 75–92, rationalize each denominator. Simplify, if possible.15----------√6 + 1

Verified step by step guidance

1. Identify the expression to rationalize: \( \frac{15}{\sqrt{6} + 1} \).
2. Multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{6} - 1 \).
3. Write the expression as: \( \frac{15(\sqrt{6} - 1)}{(\sqrt{6} + 1)(\sqrt{6} - 1)} \).
4. Simplify the denominator using the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\), where \(a = \sqrt{6}\) and \(b = 1\).
5. Simplify the numerator by distributing 15: \(15\sqrt{6} - 15\).

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 is a binomial involving a square root, you can multiply by the conjugate of that binomial.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between the two terms. For instance, the conjugate of (a + b) is (a - b). When multiplying a binomial by its conjugate, the result is a difference of squares, which eliminates the square root in the denominator, making it easier to simplify the expression.

Simplifying Radicals

Simplifying radicals involves reducing a square root to its simplest form. This can include factoring out perfect squares from under the radical sign and rewriting the expression. For example, √12 can be simplified to 2√3, as 4 is a perfect square factor of 12. This process is essential for presenting the final answer in a clear and concise manner.