Can the "Babylon Method" be used Exponents with Decimal Arguments?

I was reading about the methods that the Ancient Babylonian Civilization used for approximately calculating square roots:

• Could this method be used for approximately calculating any "root" - for instance, could we use this method to approximately calculate the cube of some number "S"?

• Could this method be used for approximately calculating an exponent with a "decimal argument"? For instance, the square root of "S" can be written as S^0.5 - Could we use this method for approximately calculating S^0.3?

Although there are now more modern ways to approximate these calculations, I am interested in learning about the limitations of these ancient methods that existed far before calculators and computers!

Thanks!

Source:

• The Babylonian square root algorithm is just a special case of Newton's method. Feb 1, 2022 at 6:33
• Why do you want an algorithm to approximate the cube of S? It's just S×S×S. If you mean the cube root, then sure, that's easy to do with Newton's method. Feb 1, 2022 at 6:38
• Thank you for your reply! I wonder if the Newton method can be used for decimal exponents? Feb 1, 2022 at 6:39

Yes. The Baylonian method can indeed be generalized easily to $$x_n=\frac{1}{k}\left((k-1)x_{n-1}+\frac{S}{x_{n-1}^{k-1}}\right)$$ to compute any $$k-th$$ root (although the convergence is very slow). Once computed the $$k-th$$ root you can then compute the $$l-th$$ power if you want to calculate $$S^{l/k}$$
• Sure $K=0.127=\frac{127}{1000}=\frac{l}{k}$ Now compute the $1000-$th root and its $127$ power. Obviously this does not generalize to irrational exponents. Feb 1, 2022 at 6:41