Question

How do you simplify `sqrt216`?

Answer 1

`sqrt(216) = 6sqrt(6)`

Explanation:

The quick version is:

`sqrt(216) = sqrt(6^2*6) = 6sqrt(6)`

How do we find out that `216 = 6^2*6` ?

One way is to split off prime factors one at a time, then recombine them.

Here's a factor tree for `216`:

`color(white)(0000)216`
`color(white)(0000)"/"color(white)(0)"\"`
`color(white)(000)2color(white)(00)108`
`color(white)(000000)"/"color(white)(0)"\"`
`color(white)(00000)2color(white)(00)54`
`color(white)(0000000)"/"color(white)(00)"\"`
`color(white)(000000)2color(white)(000)27`
`color(white)(000000000)"/"color(white)(00)"\"`
`color(white)(00000000)3color(white)(0000)9`
`color(white)(000000000000)"/"color(white)(0)"\"`
`color(white)(00000000000)3color(white)(000)3`

So we find:

`216 = 2*2*2*3*3*3=(2*3*2*3)*(2*3) = 6^2*6`

By definition:

`sqrt(6^2) = 6`

For any positive numbers `a` and `b` we have:

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

Hence:

`sqrt(216) = sqrt(6^2*6) = sqrt(6^2)sqrt(6) = 6sqrt(6)`