How do you find the square root of 17?

##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##