nth Root
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2-9-2019
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nth Root
The 
th root (or "
th radical") of a quantity 
is a value 
such that 
, and therefore is the inverse function to the taking of a power. The 
th root is denoted ![r=RadicalBox[z, n]](http://mathworld.wolfram.com/images/equations/nthRoot/Inline7.gif)
or, using power notation, 
. The special case of the square root (
) is denoted
. The case 
is known as the cube root.
The quantities for which a general function equals 0 are also called roots, or sometimes zeros.
The quantities 
such that 
are called the 
th roots of unity.
Rolle proved that any complex number has exactly 
th roots (Boyer 1968, p. 476), though some are possibly degenerate. However, since complex numbers have two square roots and three cube roots, care is needed in determining which root is under consideration. For complex numbers 
, the root of interest (generally taken as the root having smallest positive complex argument) is known as the principal root. However, for real numbers, the root of interest is usually the root that is real (when it exists).
The principal 
th root of a complex number 
can be found in the Wolfram Language as z^(1/n) or equivalently Power[z, 1/n]. When only real roots are of interest, the command Surd[x, n] which returns the real-valued 
th root for real 
odd 
and the principal
th root for nonnegative real 
and even 
can be used.
The 
th root 
of a complex number 
can be found analytically by solving the equation
 |
(1)
|
Writing the 
th power of a complex number 
in terms of its norm and phase gives
so the roots have complex modulus
 |
(4)
|
and complex argument
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