Question

rationalize the denominator and simplify:
\\(\dfrac{2\sqrt{5} + 3}{\sqrt{5} - 1}\\)

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of $\sqrt{5} - 1$ is $\sqrt{5} + 1$. So we multiply the fraction by $\frac{\sqrt{5} + 1}{\sqrt{5} + 1}$:
$$\frac{(2\sqrt{5} + 3)(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)}$$

Step2: Expand the numerator and the denominator.

First, expand the numerator using the distributive property (FOIL method):
$$(2\sqrt{5})(\sqrt{5}) + (2\sqrt{5})(1) + 3(\sqrt{5}) + 3(1) = 2\times5 + 2\sqrt{5} + 3\sqrt{5} + 3 = 10 + 5\sqrt{5} + 3 = 13 + 5\sqrt{5}$$
Then, expand the denominator using the difference of squares formula $(a - b)(a + b) = a^2 - b^2$:
$$(\sqrt{5})^2 - (1)^2 = 5 - 1 = 4$$

Step3: Write the simplified fraction.

Now we have the fraction:
$$\frac{13 + 5\sqrt{5}}{4}$$

Answer:

$\frac{13 + 5\sqrt{5}}{4}$