# How do you find the square root of 10?

Jun 26, 2016

$\sqrt{10} \approx 3.16227766016837933199$ is not simplifiable.

#### Explanation:

$10 = 2 \times 5$ has no square factors, so $\sqrt{10}$ is not simplifiable.

It is an irrational number a little greater than $3$.

In fact, since $10 = {3}^{2} + 1$ is of the form ${n}^{2} + 1$, $\sqrt{10}$ has a particularly simple continued fraction expansion:

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

We can truncate this continued fraction expansion to get rational approximations to $\sqrt{10}$

For example:

sqrt(10) ~~ [3;6] = 3+1/6 = 19/6 = 3.1bar(6)

sqrt(10) ~~ [3;6,6] = 3+1/(6+1/6) = 3+6/37 = 117/37 = 3.bar(162)

Actually:

$\sqrt{10} \approx 3.16227766016837933199$