Estimate the following square root and leave the answer as a simplified fraction √(3/20)

Steps to Solve

1. Find the approximate square root of the fraction

To estimate the square root of the fraction $\frac{3}{20}$, we can calculate it directly:

$$ \sqrt{\frac{3}{20}} = \frac{\sqrt{3}}{\sqrt{20}} $$

2. Simplify the denominator

Next, we need to simplify $\sqrt{20}$. We know that:

$$ \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} $$

So, we update the expression for the square root:

$$ \sqrt{\frac{3}{20}} = \frac{\sqrt{3}}{2\sqrt{5}} $$

3. Rationalizing the denominator

To express our result as a simplified fraction, we need to rationalize the denominator. We multiply the numerator and the denominator by $\sqrt{5}$:

$$ \frac{\sqrt{3}}{2\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{15}}{2 \cdot 5} = \frac{\sqrt{15}}{10} $$

4. Final expression

Now we have expressed the square root of the fraction $\frac{3}{20}$ in a simplified form:

$$ \sqrt{\frac{3}{20}} = \frac{\sqrt{15}}{10} $$