Why is $\sqrt {12} = 2 \sqrt 3$? Why $\sqrt {12} = 2 \sqrt 3$? It is obvious? If we considered the function $f(s) = s^2 $ it is injective on positive numbers so we obtain the conclusion. But in  the same time it is an equality between irrational numbers. Suppose that we know just to compute the square roots.

By definition, $\sqrt3$ is the positive number $s$ such that $s^2=3.$ Likewise, $\sqrt{12}$ is the positive number $t$ such that $t^2=12.$ The claim, then, is that $t=2s,$ but is it true? Well, $2s$ is a positive number, since it is a product of two positive numbers. Moreover, $$(2s)^2=2^2s^2=4s^2=4\cdot 3=12.$$ Hence, $2s=t,$ as desired.
