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typeset structureth root

Theorem. If typeset structure is a non-negative real number and typeset structure a positive integer then there is exactly one non-negative real number typeset structure such that typeset structure.

The (principal)  typeset structureth root of a non-negative real number typeset structure is the unique non-negative real number typeset structure such that typeset structure.  It is denoted by typeset structure.  When typeset structure, the typeset structureth root is called the square root (instead of typeset structure we write typeset structure), and when typeset structure, the typeset structureth root is called the cube root, etc. 1

Clearly, typeset structure, typeset structure for every positive integer typeset structure. Similarly typeset structure, and typeset structure for every real typeset structure.  

Under some circumstances we can define typeset structure also for typeset structure. If typeset structure is odd and typeset structure then we can define typeset structure.  For thus defined typeset structure one should be careful with applications of rules governing the manipulations valid for principal typeset structureth roots.

If typeset structure is a positive integer, and typeset structure are positive real numbers then

a^(1/N) b^(1/N) = (a b)^(1/N)

a^(1/N)/b^(1/N) = a/b ^(1/N)

(a^(1/N))^s = a^s^(1/N)        for    every integer    s

a^(1/N) = a^s^(1/(N s))        for every positive integer s .

For an algorithms how to compute the typeset structureth root visit .

For algorithms for computing square roots visit   . For geometric constructions of square roots by compass and straightedge visit .

To compute typeset structureth root of a non-negative real number with higher precision got to   .

One should also distinguish between the typeset structureth and a root of the equation typeset structure.

Theorem. If typeset structure is not an typeset structureth power of a rational number then typeset structure is an irrational number.

Notes

1 The symbol typeset structureprobably developed form the first letter r of the word radix. The initial letter R or r  was previously used in this sense by Paciolus, Cardano, and others.  In printed form a hook for the square root, i.e. typeset structurewithout the vinculum (=the horizontal bar over the numbers inside the radical symbol),  appears for the first time in the year 1525 in Die Coss by German mathematician Christoff Rudolff.  Another German mathematician Stifel used this sign in his Arithmetica integra in 1544.

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Štefan Porubský: Nth root.

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