Square and higher roots

Recently, and I don't remember I read it, I learned that the Egyptians had a method of calculating square roots which appealed to me: make an estimate, divide that estimate into the number, average the estimate with the result, and iterate.

For example, to work out the square root of 9, make a guess, say 4; divide 4 into 9, result 2.25; average 4 and 2.25, 3.15, and use this as the next guess. It quickly settles on 3. Even a bad guess, say 10, iterates quickly: next guess being the average of 10 and 9/10.

I couldn't prove this mathematically, but thinking about it whilst driving somewhere, I can see it visually. The result of the first guess, 4 and 2.25, comprise a rectangle with area 10; the next average shortens the long side, and lengthens the short side, so that the next shape is a stubbier rectangle with 3.15 being the longer side; and the next rectangle stubbed still. Very neat.

I then decided the method should work for cube roots, and indeed it does, with the adaption that you divide the guess twice into the number, then taken the mean of the three; visually, you have a cuboid with the correct volume, which becomes more cubical on each iteration.

Whilst I can't visualise it, you would think that it works for 4th and higher powers, since again the volume of the shape remains. Alas, when I created an Excel model to test this, whilst it does indeed work for square and cube roots, for 4th and 5th powers it oscillates around the answer, never getting into a convergence to the root, unless you make a good first guess.

There might be a modification which could be made, maybe averaging the results after a while, but that wouldn't be elegant. Also, maybe there is a higher power or powers where the method does converge rather than oscillate. I might work on these two issues at some future time.

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