Question

rationalize the denominator and simplify.
\sqrt{\frac{2}{7}}

Explanation:

Step1: Apply the property of square roots

We know that $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ (where $a\geq0$ and $b > 0$). So, $\sqrt{\frac{2}{7}}=\frac{\sqrt{2}}{\sqrt{7}}$.

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{7}$. So we have $\frac{\sqrt{2}\times\sqrt{7}}{\sqrt{7}\times\sqrt{7}}$.

Step3: Simplify the expression

We know that $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$ (for $a\geq0,b\geq0$) and $\sqrt{a}\times\sqrt{a}=a$ (for $a\geq0$). So, $\sqrt{2}\times\sqrt{7}=\sqrt{2\times7}=\sqrt{14}$ and $\sqrt{7}\times\sqrt{7} = 7$. Thus, the expression becomes $\frac{\sqrt{14}}{7}$.

Answer:

$\frac{\sqrt{14}}{7}$