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In a calculation, I've come across a relation along the lines of this: ${a}^{1/2}+{b}^{1/2}$

My presumption would be that this is somewhat related to the Pythagorean relation: ${a}^{2}+{b}^{2}$

I can understand the Pythagorean relation, but not the importance of the square root relation. Is there a hidden geometric significance to the first relation, just as there is one for the Pythagorean formula?

Note: the other side of the equation in my calculations could be anything, it's not limited to ${c}^{1/2}$. Actually, in my calculations the full relation is: ${c}={d}^{1/2}\bigg[{a}^{1/2}+{b}^{1/2}\bigg]$

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    I was doing original research. But to answer it fully without revealing my work, I was relating two bodies of mass and this relation came about because of that. It's really as simple as that; relating two masses.2012-11-07

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For positive $a$ and $b$, the generalized mean with exponent 1/2: $M_{1/2}\left(a,b\right)=\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^{2}$ is equal to $a$ if $a=b$, and lies between the geometric mean and the arithmetic mean (the "average"): $\sqrt{ab}\le\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^{2}\le\frac{a+b}{2}$.

If $a$ and $b$ are areas of squares, then $M_{1/2}\left(a,b\right)$ is the area of a square with side length equal to the average of the original side lengths, and your expression is the sum of the side lengths, but without more geometric context I don't know how to get a better geometric picture of these things.