Question

How do you find the square root of 50?

Answer 1

`sqrt50=5sqrt2`

Explanation:

`sqrt50`

= `sqrt(2xxul(5xx5))`

= `5sqrt2`

Answer 2

`sqrt(50)` can be simplified as `5sqrt(2)`

We can also find rational approximations to it.

For example:

`sqrt(50) ~~ 7 14/197 ~~ 7.071066`

Explanation:

The square root of `50` is not a whole number, or even a rational number. It is an irrational number, but you can simplify it or find rational approximations for it.

First note that

`50 = 2 xx 5 xx 5`

contains a square factor `5^2`. We can use this to simplify the square root:

`sqrt(50) = sqrt(5^2*2) = sqrt(5^2)*sqrt(2) = 5 sqrt(2)`

Apart from simplifying it algebraically, what is its numerical value?

Note that `7^2 = 49`, so `sqrt(49) = 7` and `sqrt(50)` will be slightly larger than `7`.

In fact, since `50=7^2+1`, the square root of `50` is expressible as a very regular continued fraction:

`sqrt(50) = 7+1/(14+1/(14+1/(14+1/(14+1/(14+1/(14+...))))))`

This can be written as `sqrt(50) = [7;bar(14)]` where the bar over the `14` indicates the repeating part of the continued fraction.

We can terminate the continued fraction early to give us rational approximations for `sqrt(50)`.

For example:

`sqrt(50) ~~ [7;14] = 7+1/14 = 7.0bar(714285)`

`sqrt(50) ~~ [7;14,14] = 7+1/(14+1/14) = 7+14/197 ~~ 7.071066`

In fact:

`sqrt(50) ~~ 7.071067811865475244`