Rationalize each denominator. See Example 8. √2 - √3 ———— √6 - √5

Verified step by step guidance

1. Identify the need to rationalize the denominator, which is $\sqrt{6}-\sqrt{5}$.
2. Multiply both the numerator and the denominator by the conjugate of the denominator, $\sqrt{6}+\sqrt{5}$.
3. Apply the difference of squares formula to the denominator: $(a-b)(a+b)=a^2-b^2$, where $a=\sqrt{6}$ and $b=\sqrt{5}$.
4. Simplify the denominator to $6-5$ and the numerator to $(\sqrt{2}-\sqrt{3})(\sqrt{6}+\sqrt{5})$.
5. Expand the numerator using the distributive property and simplify each term.

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 square roots, one can multiply by the conjugate of that binomial.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between two terms. For instance, the conjugate of (a + b) is (a - b). In the context of rationalizing denominators, multiplying by the conjugate helps to eliminate square roots or other irrational components, simplifying the expression. This technique is particularly useful when dealing with expressions like √6 - √5.

Properties of Square Roots

Understanding the properties of square roots is essential for simplifying expressions involving them. Key properties include that √a * √b = √(a*b) and that √a - √b can be manipulated using the difference of squares. These properties allow for the simplification of complex expressions and are crucial when performing operations like rationalizing denominators.