Question

1. \\(dfrac{-5 + 2sqrt{3}}{sqrt{14}}\\)

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{14}$:
$$\frac{-5 + 2\sqrt{3}}{\sqrt{14}}\times\frac{\sqrt{14}}{\sqrt{14}}=\frac{(-5 + 2\sqrt{3})\sqrt{14}}{14}$$

Step2: Distribute the $\sqrt{14}$ in the numerator

Using the distributive property $a(b + c)=ab+ac$, we get:
$$\frac{-5\sqrt{14}+2\sqrt{3}\times\sqrt{14}}{14}$$
Simplify $\sqrt{3}\times\sqrt{14}=\sqrt{42}$:
$$\frac{-5\sqrt{14}+2\sqrt{42}}{14}$$
We can also write it as $\frac{-5\sqrt{14}}{14}+\frac{2\sqrt{42}}{14}=\frac{-5\sqrt{14}}{14}+\frac{\sqrt{42}}{7}$ (by simplifying the second fraction: $\frac{2\sqrt{42}}{14}=\frac{\sqrt{42}}{7}$)

Answer:

$\frac{-5\sqrt{14}+2\sqrt{42}}{14}$ (or equivalent simplified forms like $\frac{-5\sqrt{14}}{14}+\frac{\sqrt{42}}{7}$)