Rationalize each denominator. See Example 8. (√2 - √3)/(√6 - √5)

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Identify the expression to rationalize: $\frac{\sqrt{2} - \sqrt{3}}{\sqrt{6} - \sqrt{5}}$.

To rationalize the denominator, multiply both numerator and denominator by the conjugate of the denominator. The conjugate of $\sqrt{6} - \sqrt{5}$ is $\sqrt{6} + \sqrt{5}$.

Multiply numerator and denominator by $\sqrt{6} + \sqrt{5}$: $\frac{(\sqrt{2} - \sqrt{3})(\sqrt{6} + \sqrt{5})}{(\sqrt{6} - \sqrt{5})(\sqrt{6} + \sqrt{5})}$.

Use the difference of squares formula for the denominator: $(a - b)(a + b) = a^2 - b^2$, so the denominator becomes \$6 - 5$.

Expand the numerator by distributing each term: $(\sqrt{2})(\sqrt{6}) + (\sqrt{2})(\sqrt{5}) - (\sqrt{3})(\sqrt{6}) - (\sqrt{3})(\sqrt{5})$, then simplify each radical product.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers, such as square roots, from the denominator of a fraction. This is done to simplify the expression and make it easier to work with, often by multiplying numerator and denominator by a conjugate or an appropriate radical.

Conjugates of Binomials

The conjugate of a binomial expression a + b is a - b, and vice versa. Multiplying a binomial by its conjugate results in a difference of squares, which eliminates the square roots in the denominator, simplifying the expression to a rational number or simpler radical form.

Difference of Squares Formula

The difference of squares formula states that (a + b)(a - b) = a² - b². This identity is crucial when rationalizing denominators involving binomials with radicals, as it helps remove the square roots by converting the product into a difference of squares, which is easier to simplify.