# How do you simplify  sqrt19?

May 16, 2016

$\sqrt{19}$ is already in simplest form.

#### Explanation:

$19$ is a prime number, so has no square factors.

As a result it is not possible to simplify $\sqrt{19}$, the positive irrational number whose square is $19$.

$\sqrt{19} \approx 4.35889894354$

The continued fraction expansion of $\sqrt{19}$ looks like this:

sqrt(19) = [4; bar(2, 1, 3, 1, 2, 8)] = 4+1/(2+1/(1+1/(3+1/(1+1/(2+1/(8+1/(2+1/(1+...))))))))

We can get an economical rational approximation for $\sqrt{19}$ by truncating just before the end of the repeating part:

sqrt(19) ~~ [4; 2, 1, 3, 1, 2] = 4+1/(2+1/(1+1/(3+1/(1+1/2)))) = 170/39 = 4.bar(358974)