What is the fastest way to solve sqroot of (16)(20)+(8)(32)?

Your method looks fast enough, but for bigger numbers (or if you are not comfortable with square roots) you may want to try something like this:

sqrt[(16.32+8.32)]=sqrt[(4.4.2.2.5+4.4.2.2.4)]=sqrt[4.4.2.2.(5+4)]=
sqrt[(4.4.2.2.3.3)]= 4.2.3=24

16x20 + 8x32 ( you should see that you can take x2 from 32 and give it to 8)

16x20 + 16x16

16x (20+16)

16x36

So you can take these two numbers out of root as:

4x6 = 24

on the same line as logitech, when it comes to big numbers, and since I don't memorize the square root of large numbers, I use a combination of muzali and logitech's methods.

for this example, I'd break out your question to:

sqrt[(4x4x4x5+4x4x4x4)]=sqrt[(4x4x4x(5+4)] (I factored out 4x4x4) =sqrt[(4.4.4.(9)]

I have a simple mind, so when i see sqrt[(4x4x4x9], I can easily figure out the square root of each. the answer is therefore, 2x2x2x3=24