In Exercises 45–54, rationalize the denominator. 13/(3+√11)

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Identify the problem: The denominator contains a square root, which makes it irrational. To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator.

Write the conjugate of the denominator. The conjugate of \(3 + \sqrt{11}\) is \(3 - \sqrt{11}\).

Multiply both the numerator and denominator by the conjugate of the denominator: \( \frac{13}{3 + \sqrt{11}} \cdot \frac{3 - \sqrt{11}}{3 - \sqrt{11}} \).

Simplify the denominator using the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 3\) and \(b = \sqrt{11}\), so the denominator becomes \(3^2 - (\sqrt{11})^2 = 9 - 11 = -2\).

Simplify the numerator by distributing \(13\) across \(3 - \sqrt{11}\), resulting in \(13(3) - 13(\sqrt{11}) = 39 - 13\sqrt{11}\). Combine the simplified numerator and denominator to get the final expression.

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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 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 of the form 'a + √b', multiplying by 'a - √b' helps to simplify the expression.

Conjugates

Conjugates are pairs of binomials that have the same terms but opposite signs, such as 'a + b' and 'a - b'. When multiplied together, they yield a difference of squares, which is a rational number. In the context of rationalizing denominators, using the conjugate of a binomial containing a square root is essential for simplifying the expression effectively.

Simplifying Radicals

Simplifying radicals involves reducing a square root expression to its simplest form. This includes factoring out perfect squares from under the radical sign and rewriting the expression. Understanding how to simplify radicals is crucial when rationalizing denominators, as it allows for clearer and more manageable expressions in the final result.