Question

1. \\(\frac{-4 - 4sqrt{2}}{sqrt{19}}\\)

Explanation:

Step1: Rationalize the Denominator

To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{19}$.
$$\frac{-4 - 4\sqrt{2}}{\sqrt{19}}\times\frac{\sqrt{19}}{\sqrt{19}}=\frac{(-4 - 4\sqrt{2})\times\sqrt{19}}{19}$$

Step2: Distribute the $\sqrt{19}$ in the Numerator

Distribute $\sqrt{19}$ to both terms in the numerator:
$$\frac{-4\sqrt{19}-4\sqrt{2}\times\sqrt{19}}{19}$$
We can also factor out - 4 from the numerator:
$$\frac{-4(\sqrt{19}+\sqrt{38})}{19}$$ (since $\sqrt{2}\times\sqrt{19}=\sqrt{38}$)

Answer:

$\frac{-4\sqrt{19}-4\sqrt{38}}{19}$ (or $\frac{-4(\sqrt{19}+\sqrt{38})}{19}$)