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Theorem. If is a non-negative real number and a positive integer then there is exactly one non-negative real number such that .
The (principal) th root of a non-negative real number is the unique non-negative real number such that . It is denoted by . When , the th root is called the square root (instead of we write ), and when , the th root is called the cube root, etc. 1
Clearly, , for every positive integer . Similarly , and for every real .
Under some circumstances we can define also for . If is odd and then we can define . For thus defined one should be careful with applications of rules governing the manipulations valid for principal th roots.
If is a positive integer, and are positive real numbers then
For an algorithms how to compute the th root visit .
For algorithms for computing square roots visit . For geometric constructions of square roots by compass and straightedge visit .
To compute th root of a non-negative real number with higher precision got to .
One should also distinguish between the th and a root of the equation .
Theorem. If is not an th power of a rational number then is an irrational number.
1 | The symbol probably 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. without 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.