Question

How do you simplify `sqrt((1/18))`?

Answer 1

`sqrt(2)/6`

Explanation:

Since there can't be a square root (a radical) in the denominator, rationalize it by multiplying both the numerator and denominator with `sqrt(2)`.

`sqrt(1/18) * sqrt(2)/sqrt(2)`

`=sqrt(2/36)`

`=1/6sqrt(2)` or `sqrt(2)/6`

You multiply by `sqrt(2)` in order to make the denominator `sqrt(36)` because `sqrt(36)=6` and so you can take it out of the square root.

Answer 2

`sqrt(2)/6`

Explanation:

It is a matter of splitting the numbers up into factors that have a root if you can. Then taking these outside of the root by applying that root.

What factors are there of 18 that we can apply a root to?
The obvious ones are 2 and 9 as `2 times 9 =18` we can take the root of 9 but not of 2. So we end up with:

`sqrt((1/18)) = sqrt(1/(2 times 9)`

This can be split so that we have:

`sqrt(1/2 times 1/9) = 1/3 sqrt(1/2)`

But convention is that you do not have a root as a denominator if you can help it.

Write as: `1/3 (sqrt(1))/( sqrt(2))` This does work. Check it on a calculator.

But `sqrt(1) =1` giving:

`1/3 times 1/(sqrt(2))`

To 'get rid' of the root in the denominator multiply by the value 1 (does not change the overall values) but write the 1 in the form of `(sqrt(2))/(sqrt(2))` giving:

`1/3 times 1/sqrt(2) times sqrt(2)/sqrt(2)`

But `sqrt(2) times sqrt(2) = 2`

So now we have:
`1/3 sqrt(2)/2 = sqrt(2)/6`

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