Question

1. \\(\frac{5\sqrt{6}}{2\sqrt{27}}\\)

Explanation:

Step1: Simplify the square root in the denominator

First, we simplify $\sqrt{27}$. We know that $27 = 9\times3$, and $\sqrt{9\times3}=\sqrt{9}\times\sqrt{3}=3\sqrt{3}$. So the expression becomes $\frac{5\sqrt{6}}{2\times3\sqrt{3}}=\frac{5\sqrt{6}}{6\sqrt{3}}$.

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{3}$. So we have $\frac{5\sqrt{6}\times\sqrt{3}}{6\sqrt{3}\times\sqrt{3}}$. Using the property $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$, the numerator becomes $5\sqrt{18}$ and the denominator becomes $6\times3 = 18$.

Step3: Simplify the numerator

Simplify $\sqrt{18}$. Since $18 = 9\times2$, $\sqrt{18}=\sqrt{9}\times\sqrt{2}=3\sqrt{2}$. So the numerator is $5\times3\sqrt{2}=15\sqrt{2}$. Now the expression is $\frac{15\sqrt{2}}{18}$.

Step4: Reduce the fraction

We can divide both the numerator and the denominator by 3. So $\frac{15\div3\sqrt{2}}{18\div3}=\frac{5\sqrt{2}}{6}$.

Answer:

$\frac{5\sqrt{2}}{6}$