Question

(\frac{8}{3 - sqrt{5}})

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator \(3 - \sqrt{5}\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(3 + \sqrt{5}\).

$$ \frac{8}{3 - \sqrt{5}} \times \frac{3 + \sqrt{5}}{3 + \sqrt{5}} $$

Step2: Multiply the numerators and denominators

Using the formula \((a - b)(a + b)=a^2 - b^2\) for the denominator, we have:

$$ \frac{8(3 + \sqrt{5})}{(3)^2 - (\sqrt{5})^2}=\frac{24 + 8\sqrt{5}}{9 - 5} $$

Step3: Simplify the denominator

Simplify the denominator \(9 - 5 = 4\).

$$ \frac{24 + 8\sqrt{5}}{4} $$

Step4: Divide each term in the numerator by 4

$$ \frac{24}{4}+\frac{8\sqrt{5}}{4}=6 + 2\sqrt{5} $$

Answer:

\(6 + 2\sqrt{5}\)