Rationalize the denominator. 117−3\frac{11}{\sqrt{7} - \sqrt{3}} 7−311

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 54

1장, 문제 54

Rationalize the denominator. $$\frac{11}{\sqrt{7}$ - $\sqrt{3}$}$

검증된 단계별 안내

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

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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.

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.

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.

Rationalize the denominator. 117−3\(\frac{11}{\sqrt{7}\) - \(\sqrt{3}\)} 7−311

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Rationalizing the Denominator

Conjugates of Binomials

Difference of Squares Formula

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