Rationalize each denominator. See Example 8. 3 ———— 4 + √5

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Identify the expression to rationalize: \(\frac{3}{4 + \sqrt{5}}\).

Recall that to rationalize a denominator containing a sum with a square root, multiply numerator and denominator by the conjugate of the denominator. The conjugate of \(4 + \sqrt{5}\) is \(4 - \sqrt{5}\).

Multiply both numerator and denominator by the conjugate: \(\frac{3}{4 + \sqrt{5}} \times \frac{4 - \sqrt{5}}{4 - \sqrt{5}}\).

Use the difference of squares formula for the denominator: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 4\) and \(b = \sqrt{5}\), so the denominator becomes \(4^2 - (\sqrt{5})^2\).

Simplify the numerator by distributing 3: \(3 \times (4 - \sqrt{5})\), and simplify the denominator using the difference of squares result.

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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 or interpret. Typically, this is achieved by multiplying the numerator and denominator by a suitable expression that removes the root from the denominator.

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Rationalizing Denominators

Conjugates of Binomials

The conjugate of a binomial expression like (a + √b) is (a - √b). Multiplying a binomial by its conjugate results in a difference of squares, which eliminates the square root terms. This property is essential for rationalizing denominators that contain sums or differences involving square roots.

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Rationalizing Denominators Using Conjugates

Difference of Squares Formula

The difference of squares formula states that (x + y)(x - y) = x² - y². This formula is used to simplify products of conjugates by removing radicals. Applying this formula when rationalizing denominators helps convert expressions with roots into rational numbers, facilitating easier computation and simplification.

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