Question

simplify.
\sqrt{\frac{10}{105}}

Explanation:

Step1: Rationalize the denominator

To simplify $\sqrt{\frac{10}{105}}$, first we can simplify the fraction inside the square root. The greatest common divisor of 10 and 105 is 5, so divide numerator and denominator by 5: $\frac{10\div5}{105\div5}=\frac{2}{21}$. Now we have $\sqrt{\frac{2}{21}}$. To rationalize the denominator, multiply numerator and denominator inside the square root by 21:
$$\sqrt{\frac{2\times21}{21\times21}}=\sqrt{\frac{42}{21^2}}$$

Step2: Simplify the square root

Using the property $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ (for $b>0$), we can split the square root:
$$\frac{\sqrt{42}}{\sqrt{21^2}}$$
Since $\sqrt{21^2} = 21$, we get:
$$\frac{\sqrt{42}}{21}$$

Answer:

$\frac{\sqrt{42}}{21}$