Question

How do you rationalize the denominator and simplify `3/sqrt12`?

Answer 1

`= sqrt12/4`

Explanation:

`3 / sqrt12`

`(3 * color(blue)(sqrt12))/(( sqrt12 * color(blue)( sqrt12))`

`= ((3sqrt12))/12`

`= (cancel3sqrt12)/cancel12`

`= sqrt12/4`

Answer 2

Bit more explanation

`= (sqrt(12))/4`

Explanation:

If you multiply a value by 1 you do not change its intrinsic value.

The thing is, 1 can come in many forms. For example`" " 3/3`.

So you can write 1 as `1=sqrt(12)/sqrt(12)`

Given:`" "3/sqrt(12)`

Multiply by 1 but in the form `1=sqrt(12)/sqrt(12)`

`" "3/sqrt(12)xxsqrt(12)/(sqrt(12))`

`=(3xxsqrt(12))/(sqrt(12)xxsqrt(12))`

`=(3sqrt(12))/12`

Divide top and bottom by 3

`=(3sqrt(12)-:3)/(12-:3)`

`=(sqrt(12))/4`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(red)("If you really wanted to be absolutely correct about the answer ")`
`color(red)("consider this:")`

`sqrt(12)` is a number that when multiplied by itself results in +12

However`sqrt(12)` could be negative. Think about:

`(-3)xx(-3) = +9 = (+3)xx(+3)`

So should the answer be:

`color(green)(+-sqrt(12)/4)`

`color(red)("No!")`

The question gives: `(sqrt(12))/4`

The absence of any sign means that it is positive. Consequently the questions single `sqrt(12)` is positive. So the solutions single `sqrt(12)` has to be positive as well.