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Ch. P - Fundamental Concepts of Algebra
Blitzer - College Algebra 8th Edition
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모든 교과서Blitzer 8th EditionCh. P - Fundamental Concepts of Algebra문제 54
1장, 문제 54

Rationalize the denominator. 117−3\(\frac{11}{\sqrt{7}\) - \(\sqrt{3}\)}

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Identify the expression to rationalize: \(\frac{11}{\sqrt{7} - \sqrt{3}}\).
Recognize that the denominator is a binomial involving square roots, so multiply numerator and denominator by the conjugate of the denominator to rationalize it. The conjugate of \(\sqrt{7} - \sqrt{3}\) is \(\sqrt{7} + \sqrt{3}\).
Multiply numerator and denominator by the conjugate: \(\frac{11}{\sqrt{7} - \sqrt{3}} \times \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} + \sqrt{3}}\).
Apply the difference of squares formula to the denominator: \((a - b)(a + b) = a^2 - b^2\). Here, \(a = \sqrt{7}\) and \(b = \sqrt{3}\), so the denominator becomes \(7 - 3\).
Write the new expression as \(\frac{11(\sqrt{7} + \sqrt{3})}{7 - 3}\) and simplify the denominator to complete the rationalization process.

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주요 개념

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

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radicals (square roots) from the denominator of a fraction. This is done to simplify the expression and make it easier to work with. For denominators with square roots, multiplying numerator and denominator by a suitable expression removes the radical.
추천 영상:
02:58
Rationalizing Denominators

Conjugates of Binomials

The conjugate of a binomial expression like (√7 − √3) is (√7 + √3). Multiplying a binomial by its conjugate uses the difference of squares formula, which eliminates the square roots in the denominator by producing a rational number.
추천 영상:
05:33
Complex Conjugates

Difference of Squares Formula

The difference of squares formula states that (a − b)(a + b) = a² − b². This property is used to simplify expressions involving conjugates, especially when rationalizing denominators containing square roots, by turning the product into a difference of perfect squares.
추천 영상:
04:14
Special Products - Square Formulas
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