Question

What is `sqrt(48)` ?

Answer 1

`4sqrt3`

Explanation:

Like Asma said you can calculate the square root on a calculator for the exact amount. But sometimes a professor will ask you to simplify. This will be very useful in the future.

`sqrt48`

We want to check if there is a number with a perfect square that is a multiple of the number in our square root. In this case, there is one.
`16*3=48`

Since `16` has a perfect square root of `4` we can solve this like this...

`sqrt48`

`sqrt(16*3)` `<---` [Move square root of `16` outside of the root]

`4sqrt3`

Answer 2

`sqrt(48) = 4sqrt(3) ~~ 18817/2716 ~~ 6.92820324`

Explanation:

Note that:

`48 = 4^2*3`

So we find:

`sqrt(48) = sqrt(4^2*3) = sqrt(4^2)sqrt(3) = 4sqrt(3)`

That is the "simplest" form.

`sqrt(48)` is an irrational number a little less than `7`, since `7^2 = 49`.

In fact, since `48=7^2-1` is in the form `n^2-1` its square root can be expressed as a continued fraction with a simple pattern:

`sqrt(48) = [6;bar(1,12)] = 6+1/(1+1/(12+1/(1+1/(12+1/(1+1/(12+..))))))`

To get rational approximations, we can truncate this continued fraction early (preferably just before a "`12`")...

For example:

`sqrt(48) ~~ 6+1/(1+1/(12+1/1)) = 97/14 = 6.9bar(285714)`

`sqrt(48) ~~ 6+1/(1+1/(12+1/(1+1/(12+1/1)))) = 1351/195 = 6.9bar(282051)`

`sqrt(48) ~~ 6+1/(1+1/(12+1/(1+1/(12+1/(1+1/(12+1/1)))))) = 18817/2716 ~~ 6.92820324`

A calculator tells me that it is closer to `6.92820323`, but that's not bad.