Question

What is (square root of [6] + 2 square root of [2])(4square root of [6] - 3 square root of 2)?

Answer 1

`12 + 5sqrt12`

Explanation:

We multiply cross-multiply, that is,
`(sqrt6 + 2sqrt2)(4sqrt6 - 3sqrt2)`
equals
`sqrt6*4sqrt6 + 2sqrt2*4sqrt6 -sqrt6*3sqrt2 - 2sqrt2*3sqrt2`

The square roots time themselves equal the number under the root, so
`4*6 +8sqrt2sqrt6 - 3sqrt6sqrt2 - 6*2`

We put `sqrt2sqrt6` in evidence:
`24 + (8-3)sqrt6sqrt2 - 12`

We can join these two roots in one, after all `sqrtxsqrty = sqrt(xy)` as long as they're not both negative. So, we get
`24 + 5sqrt12 - 12`

Finally, we just take the difference of the two constants and call it a day
`12 + 5sqrt12`