Rationalize the denominator. 75−2\frac{7}{\sqrt{5} - 2} 5−27

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 51

1장, 문제 51

Rationalize the denominator. $$\frac{7}{\sqrt{5}$ - 2}$

검증된 단계별 안내

Identify the expression to rationalize: $\frac{7}{\sqrt{5} - 2}$, where the denominator contains a radical.

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{5} - 2$ is $\sqrt{5} + 2$.

Set up the multiplication: $\frac{7}{\sqrt{5} - 2} \times \frac{\sqrt{5} + 2}{\sqrt{5} + 2}$.

Multiply the numerators: $7 \times (\sqrt{5} + 2)$, and multiply the denominators using the difference of squares formula: $(\sqrt{5})^2 - (2)^2$.

Simplify the denominator to $5 - 4$, and write the expression as $\frac{7(\sqrt{5} + 2)}{1}$, which completes 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 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.

Conjugates of Binomials

The conjugate of a binomial expression a + b is a - b, and vice versa. Multiplying a binomial by its conjugate results in a difference of squares, which helps eliminate square roots in the denominator.

Difference of Squares Formula

The difference of squares formula states that (a + b)(a - b) = a² - b². This property is used when multiplying by the conjugate to remove radicals from the denominator by turning it into a rational number.

Rationalize the denominator. 75−2\(\frac{7}{\sqrt{5}\) - 2} 5−27

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Rationalizing the Denominator

Conjugates of Binomials

Difference of Squares Formula

Rationalize the denominator. $\(\frac{7}{\sqrt{5}\) - 2}$