Question

Why can you not add `2sqrt2` and `4sqrt3` together?

Answer 1

In order to add square roots and keep them in square root form, they must have the same radicand (number under the radical). Since `2sqrt2` and `4sqrt3` have different radicands they cannot be added without the use of a calculator, which would give you a decimal number. So the answer to `2sqrt2+4sqrt3` is `2sqrt2+4sqrt3` if you want to keep it in square root form. Its like trying to add `2x+4y`. Without actual values for `x` and `y`, the answer would be `2x+4y`.

If you use a calculator, `2sqrt2+4sqrt3=9.756630355022`

Answer 2

You can add the numbers. But any attempt to write the sum as single whole number times a single root of a whole number will not work.

You could write the sum as
`2(sqrt2 + 2sqrt3)` but it's not clear that that is simpler.

You could 'irrationalize' denominators and write:
`4/sqrt2 + 12/sqrt3` but that is the opposite of simpler.

You could continue by getting a common denominator.

`(4sqrt3+12sqrt2)/sqrt6`

But none of these are simpler in any clear way.