Question

1. the expression \\(\frac{8}{3 - \sqrt{5}}\\) can be written equivalently as

(1) \\(\frac{8}{3} - 8\sqrt{5}\\) (3) \\(6 + 2\sqrt{5}\\)
(2) \\(4 - 3\sqrt{5}\\) (4) \\(\frac{5}{3} + 3\sqrt{5}\\)

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator of $\frac{8}{3 - \sqrt{5}}$, we multiply the numerator and denominator by the conjugate of the denominator, which is $3 + \sqrt{5}$.
$$\frac{8(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})}$$

Step2: Simplify the denominator

Using the difference of squares formula $(a - b)(a + b)=a^2 - b^2$, where $a = 3$ and $b=\sqrt{5}$, the denominator becomes:
$$3^2-(\sqrt{5})^2=9 - 5 = 4$$

Step3: Simplify the numerator

Multiply out the numerator:
$$8(3+\sqrt{5})=24 + 8\sqrt{5}$$

Step4: Divide numerator by denominator

Now we have $\frac{24 + 8\sqrt{5}}{4}$. Divide each term in the numerator by 4:
$$\frac{24}{4}+\frac{8\sqrt{5}}{4}=6 + 2\sqrt{5}$$

Answer:

(3) $6 + 2\sqrt{5}$