Question

What is the square root of `100000`? I got as far as `100sqrt(10)`, but what is the final answer?

Answer 1

`sqrt(10)` does not simplify further.

Explanation:

`10 = 2*5` has no more square factors, so `sqrt(10)` cannot be simplified further.

So the "final answer" may be simply `sqrt(100000) = 100sqrt(10)` unless you want a decimal approximation or an expression using a continued fraction...

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`sqrt(10)` is an irrational number. It cannot be expressed as a fraction. Its decimal representation does not terminate or recur.

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If you use a calculator, it will give you an approximation like:

`sqrt(10) ~~ 3.16227766`

Hence `sqrt(100000) = 100sqrt(10) ~~ 316.227766`

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Note that `10 = 3^2+1` is of the form `n^2+1`. As a result, it has a very regular continued fraction expansion:

`sqrt(10) = [3;bar(6)] = 3+1/(6+1/(6+1/(6+1/(6+1/(6+...)))))`

We can get rational approximations to `sqrt(10)` by truncating this continued fraction early.

For example:

`sqrt(10) ~~ [3;6] = 3+1/6 = 19/6 = 3.1bar(6)`

`sqrt(10) ~~ [3;6,6] = 3+1/(6+1/6) = 3+6/37 = 117/37 = 3.bar(162)`

Answer 2

See explanation

Explanation:

Use binomial expansion

`sqrt 10= (9+1)^(1/2)`

`=3(1+1/9)^(1/2)`

`=3(1+(1/2)(1/9)+((1/2)(1/2-1))/(2!)(1/9)^2+...)`, and upon simplification,

`=3+1/6-1/216+1/3888- ..`

The sum to four terms is `3.16229..`.

The magnitudes of the ratio of consecutive terms is more than 10.

So, easily the sum here might be correct to 5-sd, rounded. And So #

5-sd `sqrt 10 = 3.1623`.