Question

express as a fraction in simplest form with a rational denominator:
\\(\frac{-6}{-4 - \sqrt{13}}\\)

Explanation:

Step1: Identify conjugate of denominator

The denominator is $-4 - \sqrt{13}$, its conjugate is $-4 + \sqrt{13}$ (since for $a + b\sqrt{c}$, conjugate is $a - b\sqrt{c}$, here $a=-4, b = - 1, c = 13$ so conjugate is $-4 - (-1)\sqrt{13}=-4+\sqrt{13}$). Multiply numerator and denominator by this conjugate.
$$\frac{-6}{-4 - \sqrt{13}}\times\frac{-4 + \sqrt{13}}{-4 + \sqrt{13}}$$

Step2: Multiply numerators and denominators

Numerator: $(-6)\times(-4 + \sqrt{13})=24-6\sqrt{13}$
Denominator: Use difference of squares formula $(a - b)(a + b)=a^{2}-b^{2}$, here $a=-4, b=\sqrt{13}$
So $(-4)^{2}-(\sqrt{13})^{2}=16 - 13 = 3$

Step3: Simplify the fraction

Now we have $\frac{24-6\sqrt{13}}{3}$, divide each term in numerator by 3:
$\frac{24}{3}-\frac{6\sqrt{13}}{3}=8 - 2\sqrt{13}$

Answer:

$8 - 2\sqrt{13}$