Question

simplify.
\sqrt{\frac{12}{15}}

Explanation:

Step1: Simplify the fraction inside the square root

First, simplify the fraction \(\frac{12}{15}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. So, \(\frac{12\div3}{15\div3}=\frac{4}{5}\). Now we have \(\sqrt{\frac{4}{5}}\).

Step2: Use the property of square roots \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\)

Applying the property \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) (where \(a = 4\) and \(b = 5\)), we get \(\frac{\sqrt{4}}{\sqrt{5}}\). Since \(\sqrt{4}=2\), this becomes \(\frac{2}{\sqrt{5}}\).

Step3: Rationalize the denominator

To rationalize the denominator, multiply the numerator and the denominator by \(\sqrt{5}\). So, \(\frac{2\times\sqrt{5}}{\sqrt{5}\times\sqrt{5}}=\frac{2\sqrt{5}}{5}\).

Answer:

\(\frac{2\sqrt{5}}{5}\)