Is there a simpler method of calculating $\sqrt[n]{x}$?

The n-th root algorithm converges quadratically (the number of correct digits is essentially doubled each step) and uses only "simpler" operations such as addition, division, integer powers.

This method is fast, reliable, and accurate and does not require anything except basic arithmetic. You know in advance how many steps are needed to get, say 100 digits. With a decent starting value, you will need about 10 steps.

The log method requires the computation of logarithms and exponentials which is in turn done approximately. So this method has an uncertain amount of computational work associated with it and accuracy control is also uncertain.

"Simplicity" is a subjective criterion and must be balanced against reliability, accuracy, speed. Many (including me) would call the n-th root algorithm simple and elegant.

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