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Does (a+b)^0.5 equal a^.5+b^.5-(2(ab)^0.5)^0.5?


Endercreeper01

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Does (a+b)1/2 equal a1/2 + b1/2 - (2ab)1/2?

If you have ( (a+b)2)1/2, then because of the binomial theorem, then (a+b)2=a2+b2+2ab.
Inserting this into the equation, then you get (a2+b2+2ab)1/2, which gives you a+b, since a2+b2+2ab=(a+b)2. If you set x=a2 and y=b2, then you have (x+y+2x1/2y1/2)1/2, which gives you x1/2+y1/2. This is also equal to x1/2 + y1/2 + (2x1/2y1/2)1/2 - (2x1/2y1/2)1/2, since (2x1/2y1/2)1/2 and -(2x1/2y1/2)1/2 cancel out. What if you removed 2x1/2y1/2 from the square root? Then, would you get x1/2 + y1/2 - (2x1/2y1/2)1/2?

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Does (a+b)1/2 equal a1/2 + b1/2 - (2ab)1/2?

So, I'm curious why you didn't try this with a few numbers:

 

a=6, b = 9.6 --> sqrt(a+b) = 3.937003937; sqrt(a) + sqrt(b) - sqrt(2ab) = -5.145381508

a=13.3, b = 0.3 --> sqrt(a+b) = 3.687817783; sqrt(a) + sqrt(b) - sqrt(2ab) = 1.369749685

 

and so on...

 

Clearly it doesn't hold except maybe in a few special cases.

Edited by Bignose
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