How do you simplify `sqrt(108)` ?

Well, first, let's explain how you simplify a square root:

Let's say the problem is this
`root(3)(x^2xxy^3)`

Our first task is to expand all our components

`root(3)(x xx x xxy xx y xx y)`

Now we need to look at our index, the `3` in the top left corner

Whatever that number is, that's the value of the exponent we're trying to create. If it's a `sqrt(color(white)(x))`, we want an `x^2`. If it's a `root(3)(color(white)(x))`, we want a `x^3`.

So, we are looking for `x^3` inside the root, or `x xx x xx x`.
Back to our example:

Let's condense everything we can to give us a cube root

`root(3)(x xx x xx y^3)`

When we have a number that matches the index, we can take it out of the root:

`yroot(3)(x xx x)`

We can simplify that to `yroot(3)sqrt(x^2)`

`color(white)(0)`

Now, let's solve your problem:

`sqrt(108)`

expand

`sqrt(2 xx 3 xx 3 xx 3 xx 2)`

what's the index?
`2`
What can we change to something squared

`sqrt(2^2 xx 3^2 xx 3)`

Bring out the squared

That gives us `2xx3sqrt(3)` or `6sqrt(3)`

`color(white)(0)`

Let's look at the other option

`2sqrt(27)`(`sqrt(4*27)`)

`2sqrt(4*27)` equals `sqrt(108)`

`2sqrt(27)*xxsqrt(108)`

`2sqrt(3xx3xx3)xxsqrt(2xx2xx3xx3xx3)`

`2sqrt(3^2xx3)xxsqrt(2^2xx3^2xx3)`

`2xx3sqrt(3)xx6sqrt(3)`

`36sqrt(3xx3)`

`36xx3`

`108`

`108` does not equal `sqrt(108)`, so the second option cannot be correct. Hopefully this helps!

When `a >=0` and/or `b >= 0` then:

`sqrt(ab) = sqrt(a)sqrt(b)`

Also, if `a >= 0` then:

`sqrt(a^2) = a`

The prime factorisation of `108` is:

`108 = 2*2*3*3*3`

So we find:

`sqrt(108) = sqrt(2^2*27) = sqrt(2^2)sqrt(27) = 2sqrt(27)`

`sqrt(108) = sqrt(3^2*12) = sqrt(3^2)sqrt(12) = 3sqrt(12)`

`sqrt(108) = sqrt(6^2*3) = sqrt(6^2)sqrt(3) = 6sqrt(3)`

So there are several "simplifications" of `sqrt(108)`.

When asked to simplify such a square root, what is normally expected is the expression that no longer has any square factors in the radicand. So in our example, `6sqrt(3)` is considered simplified, whereas `2sqrt(27)` or `3sqrt(12)` are not fully simplified.

So if your teacher marked `2sqrt(27)` (from `sqrt(108) = sqrt(4*27)`) as being wrong, it was not because it was not equal (it is), but because it was not fully simplified.