# How do you simplify sqrt10?

Feb 17, 2016

It is not possible to simplify $\sqrt{10}$, but we can find rational approximations quite easily...

#### Explanation:

$10 = 2 \cdot 5$ has no square factors, so $\sqrt{10}$ has no simpler form.

It is an irrational number, that is it not expressible as $\frac{p}{q}$ for integers $p$ and $q$. Neither will its decimal expansion repeat or terminate.

$10 = {3}^{2} + 1$, hence $\sqrt{10}$ is a little more than $3$.

In fact, it can be expressed as a continued fraction:

sqrt(10) = [3;bar(6)] = 3 + 1/(6+1/(6+1/(6+1/(6+...))))

We can get rational approximations for $\sqrt{10}$ by terminating this continued fraction.

For example:

$\sqrt{10} \approx 3 + \frac{1}{6 + \frac{1}{6}} = \frac{117}{37} \approx 3. \overline{1} 6 \overline{2}$