Question

1. \\(\frac{4sqrt{3} - sqrt{5}}{sqrt{2}}\\)

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{2}$. So we have $\frac{(4\sqrt{3} - \sqrt{5})\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}$.

Step2: Simplify the denominator and numerator

The denominator $\sqrt{2}\times\sqrt{2} = 2$. For the numerator, we use the distributive property: $(4\sqrt{3})\times\sqrt{2}-\sqrt{5}\times\sqrt{2}=4\sqrt{6}-\sqrt{10}$. So the expression becomes $\frac{4\sqrt{6}-\sqrt{10}}{2}$.

Step3: Split the fraction (optional)

We can split the fraction as $\frac{4\sqrt{6}}{2}-\frac{\sqrt{10}}{2}=2\sqrt{6}-\frac{\sqrt{10}}{2}$.

Answer:

$2\sqrt{6}-\frac{\sqrt{10}}{2}$ (or $\frac{4\sqrt{6}-\sqrt{10}}{2}$)