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

Aug 26, 2016

$\sqrt{10}$ does not simplify further.

#### Explanation:

$10 = 2 \cdot 5$ has no more square factors, so $\sqrt{10}$ cannot be simplified further.

So the "final answer" may be simply $\sqrt{100000} = 100 \sqrt{10}$ 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:

$\sqrt{10} \approx 3.16227766$

Hence $\sqrt{100000} = 100 \sqrt{10} \approx 316.227766$

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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)

Aug 26, 2016

See explanation

#### Explanation:

Use binomial expansion

$\sqrt{10} = {\left(9 + 1\right)}^{\frac{1}{2}}$

$= 3 {\left(1 + \frac{1}{9}\right)}^{\frac{1}{2}}$

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

$= 3 + \frac{1}{6} - \frac{1}{216} + \frac{1}{3888} - . .$

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 $\sqrt{10} = 3.1623$.