Square roots manipulation

[mucpa2011]

Please see the attached square roots problem. I can't seem to follow the solution.

My method was to first multiply the 8 and the 32, so that I could get 256 (a nice, perfect square of 16).

Then I simplified the first expression under the radical into 4 square roots of 20, leaving me with an incorrect answer of:

4 square roots of 20 + 16.

I don't understand the simplification in the text. I understand that (16)(16) is equivalent to (8)(32), but after that first second simplification, I am lost.

Could anyone elaborate on why the method I used above is wrong, and how the text was able to get rid of one of those (16)'s in the second operation?

 

[tkhunny]

Maybe this version will help you see it.

\(\displaystyle \sqrt{16*20 + 8*32} = \sqrt{4*(4*20 + 2*32)} = 2\sqrt{4*20 + 2*32}\)

\(\displaystyle 2\sqrt{4*20 + 2*32} = 2\sqrt{4*(20 + 2*8)} = 4\sqrt{20 + 2*8}\)

\(\displaystyle 4\sqrt{20 + 2*8} = 4\sqrt{4*(5 + 2*2)} = 8\sqrt{5+2*2}\)

\(\displaystyle 8\sqrt{5+2*2} = 8\sqrt{5+4} = 8\sqrt{9} = 8*3 = 24\)

Just one piece at a time. No one should care what order you do things, so long as you find all the perfect squares hiding in there.