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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.