Question

How do you simplify `sqrt(442)`?

Answer 1

`sqrt(442)` cannot be simplified, but it has a simple continued fraction expansion:

`sqrt(442) = [21;bar(42)] = 21 + 1/(42+1/(42+1/(42+...)))`

Explanation:

`442 = 2*13*17` has no square factors, so we cannot simplify the square root.

It is an irrational number, so it cannot be expressed in the form `p/q` for any integers `p` and `q`. Neither will its decimal expansion terminate or repeat.

However, it is of the form `n^2+1`, since `442 = 441+1 = 21^2+1`.

As a result, the continued fraction for its square root takes a particularly simple form:

`sqrt(442) = [21;bar(42)] = 21 + 1/(42+1/(42+1/(42+...)))`

In general, any positive integer of the form `n^2+1` has square root:

`sqrt(n^2+1) = [n;bar(2n)]`