# What is the simplest radical form of sqrt(5) / sqrt(6)?

Jul 29, 2015

$\frac{\sqrt{5}}{\sqrt{6}} = \sqrt{\frac{5}{6}} = \sqrt{0.8333 \ldots}$

#### Explanation:

When dealing with positive numbers $p$ and $q$, it's easy to prove that
$\sqrt{p} \cdot \sqrt{q} = \sqrt{p \cdot q}$
$\frac{\sqrt{p}}{\sqrt{q}} = \sqrt{\frac{p}{q}}$

For instance, the latter can be proven by squaring the left part:
${\left(\frac{\sqrt{p}}{\sqrt{q}}\right)}^{2} = \frac{\sqrt{p} \cdot \sqrt{p}}{\sqrt{q} \cdot \sqrt{q}} = \frac{p}{q}$
Therefore, by definition of a square root,
from
$\frac{p}{q} = {\left(\frac{\sqrt{p}}{\sqrt{q}}\right)}^{2}$
follows
$\sqrt{\frac{p}{q}} = \frac{\sqrt{p}}{\sqrt{q}}$

Using this, the expression above can be simplified as
$\frac{\sqrt{5}}{\sqrt{6}} = \sqrt{\frac{5}{6}} = \sqrt{0.8333 \ldots}$