I was reading this article about square root and they simplify $\sqrt{75}$ to $5\sqrt{3}$. Is there a way to ensure that the answer is correct, going from $5\sqrt{3}$ to $\sqrt{75}$?
For example, I was doing this exercise:
$\sqrt{456}$
I factored and simplified it out, so I ended up with $2\sqrt{114}$, how could I ensure it's the right answer?
There are plenty of ways to do it depending on what tools you have available.
The simplest option is to square both answers and check that you get the same thing, for example:
$$\begin{eqnarray}(2\sqrt{114})^2 & = & 2^2 \times (\sqrt{114})^2 \\ & = & 4 \times 114 \\ & = & 456 \\ & = & (\sqrt{456})^2\end{eqnarray}$$
If you have a calculator on hand you can also just plug both of them in and see if the answers look the same - this will be better for catching errors of scale (e.g. if you accidentally wrote $\sqrt{456} = 4 \sqrt{114}$ rather than $2 \sqrt{114}$) compared to transcription errors (e.g. if you just look at the first few decimal places, $\sqrt{1001} \approx \sqrt{1002}$ so you might not spot that they're different).