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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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.

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Rationalizing Denominators

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.

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Rationalizing Denominators Using Conjugates

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.

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