Rationalize each denominator. See Example 8. (√3 + 1)/(1 - √3)

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Identify the expression to rationalize: \(\frac{\sqrt{3} + 1}{1 - \sqrt{3}}\).

Recognize that the denominator is a binomial involving a square root, so multiply numerator and denominator by the conjugate of the denominator to rationalize it. The conjugate of \(1 - \sqrt{3}\) is \(1 + \sqrt{3}\).

Multiply both numerator and denominator by \(1 + \sqrt{3}\): \(\frac{\sqrt{3} + 1}{1 - \sqrt{3}} \times \frac{1 + \sqrt{3}}{1 + \sqrt{3}}\).

Use the difference of squares formula for the denominator: \((1 - \sqrt{3})(1 + \sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3\).

Expand the numerator by distributing: \((\sqrt{3} + 1)(1 + \sqrt{3}) = \sqrt{3} \times 1 + \sqrt{3} \times \sqrt{3} + 1 \times 1 + 1 \times \sqrt{3}\).

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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 root terms by converting the product into a difference of squares.

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