nth Root - Computing Principal Roots

Computing Principal Roots

The nth root of an integer is not always an integer, and if it is not an integer then it is not a rational number. For instance, the fifth root of 34 is

where the dots signify that the decimal expression does not end after any finite number of digits. Since in this example the digits after the decimal never enter a repeating pattern, the number is irrational.

The nth root of a number A can be computed by the nth root algorithm, a special case of Newton's method. Start with an initial guess x₀ and then iterate using the recurrence relation

until the desired precision is reached.

Depending on the application, it may be enough to use only the first Newton approximant:

For example, to find the fifth root of 34, note that 25 = 32 and thus take x = 32 and y = 2 in the above formula. This yields

The error in the approximation is only about 0.03%.

Newton's method can be modified to produce a generalized continued fraction for the nth root which can be modified in various ways as described in that article. For example:

$\sqrt{z} = \sqrt{x^n+y} = x+\cfrac{y} {nx^{n-1}+\cfrac{(n-1)y} {2x+\cfrac{(n+1)y} {3nx^{n-1}+\cfrac{(2n-1)y} {2x+\cfrac{(2n+1)y} {5nx^{n-1}+\cfrac{(3n-1)y} {2x+\ddots}}}}}};$

$\sqrt{z}=x+\cfrac{2x\cdot y}{n(2z - y)-y-\cfrac{(1^2n^2-1)y^2}{3n(2z - y)-\cfrac{(2^2n^2-1)y^2}{5n(2z - y)-\cfrac{(3^2n^2-1)y^2}{7n(2z - y)-\ddots}}}}.$

In the case of the fifth root of 34 above (after dividing out selected common factors):

$\sqrt{34} = 2+\cfrac{1} {40+\cfrac{4} {4+\cfrac{6} {120+\cfrac{9} {4+\cfrac{11} {200+\cfrac{14} {4+\ddots}}}}}} =2+\cfrac{4\cdot 1}{165-1-\cfrac{4\cdot 6}{495-\cfrac{9\cdot 11}{825-\cfrac{14\cdot 16}{1155-\ddots}}}}.$

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