In Exercises 75–92, rationalize each denominator. Simplify, if possible. 2√6 + √5--------------3√6 - √5

Verified step by step guidance

1. Identify the need to rationalize the denominator by eliminating the square roots.
2. Multiply both the numerator and the denominator by the conjugate of the denominator, which is \(3\sqrt{6} + \sqrt{5}\).
3. Apply the distributive property (FOIL method) to both the numerator and the denominator.
4. In the denominator, use the difference of squares formula: \((a-b)(a+b) = a^2 - b^2\).
5. Simplify the expression by combining like terms and simplifying any square roots.

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, multiplying by the same square root can help achieve this.

Simplifying Radicals

Simplifying radicals refers to the process of reducing a square root or other root to its simplest form. This involves factoring out perfect squares from under the radical sign and rewriting the expression. For instance, √12 can be simplified to 2√3, as 12 = 4 × 3, and 4 is a perfect square.

Combining Like Terms

Combining like terms is a fundamental algebraic skill that involves adding or subtracting terms that have the same variable raised to the same power. In the context of rational expressions, this means simplifying the numerator or denominator by merging terms that share the same radical or numerical component, which can lead to a more concise expression.