Simplify. See Example 9. (1/2)/(1 - (√5/2))

검증된 단계별 안내

Rewrite the given expression clearly as a fraction: \(\frac{\frac{1}{2}}{1 - \frac{\sqrt{5}}{2}}\).

To simplify the complex fraction, multiply the numerator and denominator by 2 to eliminate the fractions inside the denominator: \(\frac{1}{2} \div \left(1 - \frac{\sqrt{5}}{2}\right) = \frac{1}{2} \times \frac{2}{2 - \sqrt{5}}\).

Simplify the multiplication in the numerator: \(\frac{1}{2} \times 2 = 1\), so the expression becomes \(\frac{1}{2 - \sqrt{5}}\).

Rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator, which is \(2 + \sqrt{5}\): \(\frac{1}{2 - \sqrt{5}} \times \frac{2 + \sqrt{5}}{2 + \sqrt{5}}\).

Use the difference of squares formula to simplify the denominator: \((2)^2 - (\sqrt{5})^2 = 4 - 5 = -1\). Then write the simplified expression as \(\frac{2 + \sqrt{5}}{-1}\) and simplify the negative sign.

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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 typically done by multiplying the numerator and denominator by a conjugate or an appropriate radical to create a rational denominator, simplifying the expression.

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 the difference of squares, which removes the square root terms and simplifies the expression, a key step in rationalizing denominators.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing square roots to their simplest form by factoring out perfect squares. This process helps in making expressions easier to work with and is essential when performing operations like addition, subtraction, or rationalization.