# Square root of 5/8

#### Understand the Problem

The question is asking for the square root of the fraction 5/8, which involves understanding how to calculate square roots of rational numbers.

$\frac{\sqrt{10}}{4}$

The square root of the fraction $\frac{5}{8}$ is $\frac{\sqrt{10}}{4}$.

#### Steps to Solve

1. Identify the square root of a fraction

To find the square root of a fraction, you can take the square root of the numerator and the denominator separately. So for the fraction $\frac{5}{8}$, we calculate: $$\sqrt{\frac{5}{8}} = \frac{\sqrt{5}}{\sqrt{8}}$$

1. Simplify the square root of the denominator

Next, we can simplify $\sqrt{8}$. We know that $8$ can be expressed as $4 \cdot 2$, and we can take the square root of $4$ out of the square root: $$\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$$

1. Substitute back in

Now we can substitute our simplification back into the equation: $$\sqrt{\frac{5}{8}} = \frac{\sqrt{5}}{2\sqrt{2}}$$

1. Rationalize the denominator

To make the denominator a rational number, multiply the numerator and the denominator by $\sqrt{2}$: $$\frac{\sqrt{5}}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{10}}{2 \cdot 2} = \frac{\sqrt{10}}{4}$$

The square root of the fraction $\frac{5}{8}$ is $\frac{\sqrt{10}}{4}$.