Question

What is the square root of `756487` minus the square of `7654` ?

Answer 1

`sqrt(756487)-58583716`

Explanation:

The prime factorisation of `756487` is:

`756487 = 277*2731`

Since this has no square factors, the square root cannot be simplified.

`7654^2 = 7654*7654 = 58583716`

So:

`sqrt(756487)-7654^2 = sqrt(756487)-58583716`

Footnote

In common with any square root, `sqrt(756487)` has a repeating continued fraction. In the case of `756487` it takes a while before it repeats:

`sqrt(756487) =`

`[869;bar(1,3,4,1,2,2,1,1,579,3,1,14,8,1,1,192,1,3,44,2,1,5,64,)`
`bar(3,1,133,17,21,2,2,1,1,44,51,7,7,5,14,1,2,16,1,2,2,21,1,1,2,4,1,)`
`bar(1,3,1,4,1,9,2,6,1,6,2,1,5,2,1,1,1,3,3,6,2,3,2,8,1,1,7,1,10,1,)`
`bar(19,3,4,1,2,5,1,1,1,34,1,5,1,3,2,1,4,2,1,1,1,4,11,1,1,1,1,1,1,2,)`
`bar(3,1,1,1,7,1,13,3,1,6,1,5,3,7,1,1,1,41,1,3,2,3,1,1,3,2,2,4,1,1,)`
`bar(13,1,1,2,4,2,1,11,4,2,5,4,1,1,7,1,2,4,3,1,2,1,6,1,1,2,1,5,1,1,)`
`bar(1,8,1,2,2,1,2,1,1,1,1,6,18,1,27,9,5,1,19,6,3,9,27,1,1,59,2,)`
`bar(9,3,82,1,1,19,2,28,1,247,1,1,6,6,9,1,1,1,82,5,1,1,2,1,1,2,)`
`bar(1,2,1,1,4,27,2,1,1,5,6,5,5,1,2,1,8,2,6,1,1,18,1,1,2,1,1,1,2,1,)`
`bar(1,2,2,20,1,1,5,1,6,7,6,7,6,1,5,1,1,20,2,2,1,1,2,1,1,1,2,1,1,)`
`bar(18,1,1,6,2,8,1,2,1,5,5,6,5,1,1,2,27,4,1,1,2,1,2,1,1,2,1,1,5,)`
`bar(82,1,1,1,9,6,6,1,1,247,1,28,2,19,1,1,82,3,9,2,59,1,1,27,)`
`bar(9,3,6,19,1,5,9,27,1,18,6,1,1,1,1,2,1,2,2,1,8,1,1,1,5,1,2,1,1,6,)`
`bar(1,2,1,3,4,2,1,7,1,1,4,5,2,4,11,1,2,4,2,1,1,13,1,1,4,2,2,3,1,1,)`
`bar(3,2,3,1,41,1,1,1,7,3,5,1,6,1,3,13,1,7,1,1,1,3,2,1,1,1,1,1,1,11,)`
`bar(4,1,1,1,2,4,1,2,3,1,5,1,34,1,1,1,5,2,1,4,3,19,1,10,1,7,1,1,8,)`
`bar(2,3,2,6,3,3,1,1,1,2,5,1,2,6,1,6,2,9,1,4,1,3,1,1,4,2,1,1,21,)`
`bar(2,2,1,16,2,1,14,5,7,7,51,44,1,1,2,2,21,17,133,1,3,64,5,1,)`
`bar(2,44,3,1,192,1,1,8,14,1,3,579,1,1,2,2,1,4,3,1,1738)]`

`=869+1/(1+1/(3+1/(4+1/(1+1/(2+1/(2+1/(1+1/(1+1/(579+...))))))))`

Notice the large term `579`. That means that we can get a good rational approximation for `sqrt(756487)` by truncating just before it:

`sqrt(756487) ~~ 869+1/(1+1/(3+1/(4+1/(1+1/(2+1/(2+1/(1+1/1))))))) = 293110/337 ~~ 869.76261127596`

Actually:

`sqrt(756487) ~~ 869.76261129115`