# Geometric Significance of the Addition of Square Roots of Two Numbers

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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Without knowing the details, that is probably more geometrically meaningful written as $c=\sqrt{ad}+\sqrt{bd}$. – Henning Makholm Oct 14 '12 at 14:59
where did you come across this relation? – john mangual Nov 7 '12 at 17:49
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. – Zchpyvr Nov 7 '12 at 23:29

## 1 Answer

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.

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