How do you find the square root of 17?

1 Answer
Jun 26, 2016

Answer:

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

We can calculate rational approximations like:

#sqrt(17) ~~ 268/65 ~~ 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) ~~ 4.12310562561766054982#