# How do you find the square root of 17?

Jun 26, 2016

$\sqrt{17}$ is not simplifiable and is irrational.

We can calculate rational approximations like:

$\sqrt{17} \approx \frac{268}{65} \approx 4.1231$

#### Explanation:

Since $17$ is prime, it has no square factors, so $\sqrt{17}$ cannot be simplified.

It is an irrational number a little larger than $4$.

Since $17 = {4}^{2} + 1$ is in the form ${n}^{2} + 1$, $\sqrt{17}$ has a particularly simple continued fraction expansion:

sqrt(17) = [4;bar(8)] = 4+1/(8+1/(8+1/(8+1/(8+1/(8+1/(8+...))))))

You can terminate this continued fraction expansion early to get rational approximations to $\sqrt{17}$.

For example:

sqrt(17) ~~ [4;8,8] = 4+1/(8+1/8) = 4+8/65 = 268/65 = 4.1bar(230769)

Actually:

$\sqrt{17} \approx 4.12310562561766054982$