Question

What is `sqrt65` in simplified radical form?

Answer 1

`sqrt(65)` cannot be simplified.

Explanation:

The prime factorisation of `65` is:

`65 = 5 * 13`

Since this has no square factors `sqrt(65)` cannot be simplified.

Bonus

`65 = 64+1 = 8^2+1`

is in the form `n^2+1`.

The square roots of such numbers have a simple form of continued fraction expansion:

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

So in our example:

`sqrt(65) = [8;bar(16)] = 8+1/(16+1/(16+1/(16+1/(16+1/(16+1/(16+...))))))`

You can get rational approximations of `sqrt(65)` to any desired accuracy by truncating the continued fraction expansion early.

For example:

`sqrt(65) ~~ [8;16] = 8+1/16 = 8.0625`

`sqrt(65) ~~ [8;16,16] = 8+1/(16+1/16) = 8+16/257 ~~ 8.0622568`

The actual value is more like:

`sqrt(65) ~~ 8.0622577483`