Question

rationalize the denominator.
\\(\frac{8}{1 - \sqrt{5}}\\)
\\(\frac{8}{1 - \sqrt{5}} = \square\\)
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Identify conjugate of denominator

The denominator is \(1 - \sqrt{5}\), its conjugate is \(1 + \sqrt{5}\). Multiply numerator and denominator by this conjugate.

$$ \frac{8}{1 - \sqrt{5}} \times \frac{1 + \sqrt{5}}{1 + \sqrt{5}} $$

Step2: Multiply numerators and denominators

Numerator: \(8\times(1 + \sqrt{5}) = 8 + 8\sqrt{5}\)
Denominator: Use difference of squares \((a - b)(a + b)=a^2 - b^2\), so \((1)^2 - (\sqrt{5})^2 = 1 - 5=-4\)
Now the fraction is \(\frac{8 + 8\sqrt{5}}{-4}\)

Step3: Simplify the fraction

Divide each term in numerator by \(-4\):
\(\frac{8}{-4}+\frac{8\sqrt{5}}{-4}=-2 - 2\sqrt{5}\) (or factor out \(-2\): \(-2(1 + \sqrt{5})\))

Answer:

\(-2 - 2\sqrt{5}\) (or \(-2(1 + \sqrt{5})\))