Question

How do you simplify `sqrt(1414)`?

Answer 1

`sqrt(1414)` is already in simplest form.

Explanation:

The prime factorisation of `1414` is:

`1414 = 2*7*101`

This contains no square factors, so the square root is already in simplest form.

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Notes

Should the `1414` in the question have been `1444`?

If it was, then we would find:

`1444 = 2*2*19*19 = 38^2`

So:

`sqrt(1444) = 38`

We also find that:

`37^2 = 1369 < 1414 < 1444 = 38^2`

So `sqrt(1414)` is an irrational number somewhere between `37` and `38`.

If you would like a rational approximation, we can start by linearly interpolating between `37` and `38`, finding:

`sqrt(1414) ~~ 37+(1414-1369)/(1444-1369) = 37+45/75 = 37+3/5 = 188/5`

This will be slightly less than `sqrt(1414)`.

We find:

`(188/5^2) = 35344/25 = 35350/25-6/25 = 1414-6/25`

That's not bad, but if we want greater accuracy, we can use a generalised continued fraction based on this approximation.

In general we have:

`sqrt(a^2+b) = a+b/(2a+b/(2a+b/(2a+b/(2a+...))))`

Putting `a=188/5` and `b=6/25` we get:

`sqrt(1414) = 188/5 + (6/25)/(376/5+(6/25)/(376/5+(6/25)/(376/5+...)))`

You can terminate this continued fraction to get rational approximations, such as:

`sqrt(1414) ~~ 188/5 + (6/25)/(376/5+(6/25)/(376/5)) = 13291036/353455 ~~ 37.603191353921`

A calculator tells me that `sqrt(1414)` is a little closer to:

`37.60319135392633134161`