Question

How do you simplify `sqrt85`?

Answer 1

`sqrt(85)` is not simplifiable.

Explanation:

The prime factorisation of `85` is:

`85 = 5 * 17`

This contains no square factors, so `sqrt(85)` is not simplifiable.

If you wish, you can factor it as:

`sqrt(85) = sqrt(5)sqrt(17)`

but I don't think that counts as simplification.

The general idea is that if a radicand contains square factors then it can be simplified. For example:

`sqrt(24) = sqrt(2^2*6) = sqrt(2^2)sqrt(6) = 2sqrt(6)`

If it has no square factors then this kind of simplification is not possible.

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Random Bonus

Any square root is expressible as a repeating continued fraction. in our example:

`sqrt(85) = [9;bar(4,1,1,4,18)]`

`=9+1/(4+1/(1+1/(1+1/(4+1/(18+1/(4+1/(1+1/(1+1/(4+1/(18+...))))))))))`

You can truncate the continued fraction to give you rational approximations for the square root.

For example:

`sqrt(85) ~~ [9;4] = 9+1/4 = 37/4 = 9.25`

`sqrt(85) ~~ [9;4,1,1,4] = 9+1/(4+1/(1+1/(1+1/(4)))) = 378/41 = 9.bar(21951)`

Actually `sqrt(85) ~~ 9.219544457`