Question

1. \\(\frac{10}{3sqrt{20}}\\)

Explanation:

Step1: Simplify the square root

First, we simplify \(\sqrt{20}\). We know that \(20 = 4\times5\), and \(\sqrt{4\times5}=\sqrt{4}\times\sqrt{5} = 2\sqrt{5}\) (using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a\geq0,b\geq0\)). So the expression becomes \(\frac{10}{3\times2\sqrt{5}}\).

Step2: Simplify the denominator

Now, calculate \(3\times2 = 6\) in the denominator. So we have \(\frac{10}{6\sqrt{5}}\). We can simplify the fraction \(\frac{10}{6}\) by dividing both the numerator and the denominator by 2, which gives \(\frac{5}{3\sqrt{5}}\).

Step3: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{5}\). So we get \(\frac{5\times\sqrt{5}}{3\sqrt{5}\times\sqrt{5}}\).

Step4: Simplify the denominator after rationalization

We know that \(\sqrt{5}\times\sqrt{5}=5\) (using the property \(\sqrt{a}\times\sqrt{a}=a\) for \(a\geq0\)). So the denominator becomes \(3\times5 = 15\), and the numerator becomes \(5\sqrt{5}\). So the expression is \(\frac{5\sqrt{5}}{15}\).

Step5: Simplify the final fraction

Now, we simplify \(\frac{5\sqrt{5}}{15}\) by dividing both the numerator and the denominator by 5. This gives \(\frac{\sqrt{5}}{3}\).

Answer:

\(\frac{\sqrt{5}}{3}\)