Basic Square Root Question

I was reading an article on Purplemath about simplifying the square root terms. For instance in $\displaystyle \sqrt{144}$ they instruct users to break down as a product and simply from there $\displaystyle \sqrt{144}=\sqrt{9*16}=3*4=12$.

However what if you used a different pair of factors? Such as $\displaystyle \sqrt{144}=\sqrt{3*48}$? I understand once you get a perfect square you can remove it from under the radicand, but I attempted this problem and could not figure it out with the pair of factors I have. $\displaystyle \sqrt{144}=\sqrt{3*48}=\sqrt{3}\sqrt{2x24}=\sqrt{3 }\sqrt{2}\sqrt{2x12}$ etc

I understand that we could simply solve by

knowing $\displaystyle 12^2=144$ but that defeats the purpose of this exercise.