I need to show that $a+b\sqrt{2}$ is associative over multiplication. This is what I have so far. I may be taking a wrong route so just please let me know.
$(a+b\sqrt{2})*((c+d \sqrt{2})*(e+f \sqrt{2} )) = \\ (a+b\sqrt{2})*(ce+2df+(cf+de)\sqrt{2}) $
and then I just opened up the brackets and got an ugly looking thing that I am not sure what to do next. $(ace+2adf+(acf+ade)\sqrt{2}+bce\sqrt{2}+2bdf\sqrt{2}+2(bcf+bde)$
How would I go from here?