How do you find the square root of 17?

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Answer 1 · 2016-06-26T18:10:11

$sqrt(17)$ is not simplifiable and is irrational.

We can calculate rational approximations like:

$sqrt(17) ~~ 268/65 ~~ 4.1231$

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:

$sqrt(17) ~~ 4.12310562561766054982$