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

Apr 22, 2015

In order to add square roots and keep them in square root form, they must have the same radicand (number under the radical). Since $2 \sqrt{2}$ and $4 \sqrt{3}$ have different radicands they cannot be added without the use of a calculator, which would give you a decimal number. So the answer to $2 \sqrt{2} + 4 \sqrt{3}$ is $2 \sqrt{2} + 4 \sqrt{3}$ if you want to keep it in square root form. Its like trying to add $2 x + 4 y$. Without actual values for $x$ and $y$, the answer would be $2 x + 4 y$.

If you use a calculator, $2 \sqrt{2} + 4 \sqrt{3} = 9.756630355022$

Apr 22, 2015

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
$2 \left(\sqrt{2} + 2 \sqrt{3}\right)$ but it's not clear that that is simpler.

You could 'irrationalize' denominators and write:
$\frac{4}{\sqrt{2}} + \frac{12}{\sqrt{3}}$ but that is the opposite of simpler.

You could continue by getting a common denominator.

$\frac{4 \sqrt{3} + 12 \sqrt{2}}{\sqrt{6}}$

But none of these are simpler in any clear way.