Question

4 find the reduced radical form of \\(\frac{3}{1 - \sqrt{5}}\\)

Explanation:

Step1: Rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator, which is \(1 + \sqrt{5}\).

\[
\frac{3}{1 - \sqrt{5}} \times \frac{1 + \sqrt{5}}{1 + \sqrt{5}}
\]

Step2: Multiply the numerators and the denominators. For the numerator: \(3\times(1 + \sqrt{5}) = 3 + 3\sqrt{5}\). For the denominator, use the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), so \((1 - \sqrt{5})(1 + \sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5=-4\).

\[
\frac{3 + 3\sqrt{5}}{-4}
\]

Step3: Simplify the fraction by factoring out -1 from the numerator (or just distributing the negative sign).

\[
-\frac{3 + 3\sqrt{5}}{4}=\frac{-3 - 3\sqrt{5}}{4}
\]

Answer:

\(\frac{-3 - 3\sqrt{5}}{4}\) (or equivalently \(-\frac{3 + 3\sqrt{5}}{4}\))