Question

1. \\(\frac{3\sqrt{5} - 3\sqrt{3}}{\sqrt{18}}\\)

Explanation:

Step1: Simplify the denominator

Simplify $\sqrt{18}$. We know that $18 = 9\times2$, so $\sqrt{18}=\sqrt{9\times2}=\sqrt{9}\times\sqrt{2} = 3\sqrt{2}$. Now the expression becomes $\frac{3\sqrt{5}-3\sqrt{3}}{3\sqrt{2}}$.

Step2: Factor out the common term in the numerator

Factor out $3$ from the numerator: $\frac{3(\sqrt{5}-\sqrt{3})}{3\sqrt{2}}$. The $3$ in the numerator and denominator cancels out, leaving $\frac{\sqrt{5}-\sqrt{3}}{\sqrt{2}}$.

Step3: Rationalize the denominator

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{2}$: $\frac{(\sqrt{5}-\sqrt{3})\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{10}-\sqrt{6}}{2}$.

Answer:

$\frac{\sqrt{10}-\sqrt{6}}{2}$