An nth root undoes raising a number to the nth power
If n copies of r multiply to x, then r is an nth root of x. Degree two is a square root; degree three a cube root; higher degrees take ordinal names. Extracting roots is the inverse craft of exponentiation, often written with radicals or fractional powers.
Here n is a positive integer called the index or degree, and x is the radicand. The familiar radical sign usually hides the index two for square roots, while cube roots and beyond keep their numbers visible. For fixed n, root extraction reverses the map that sends a base to its nth power, which is why fractional exponents such as x^(1/n) mean the same thing as radical notation.
Roots are rarely unique. Both 3 and −3 square to 9, while a negative real number has no real square root at all, only two imaginary ones. Every nonzero complex number has exactly n distinct nth roots, spaced evenly around a circle of fixed size; for zero the circle shrinks to a single point. By convention the principal root is the one with the largest real part, so a positive number's principal root is positive. The nth roots of 1, the roots of unity, are central to number theory, the theory of equations and the Fourier transform.
An unresolved root is called a surd or radical. The word goes back to al-Khwarizmi around 825, who called rational numbers audible and irrational ones inaudible; the Arabic for deaf became Latin surdus, used by Gerard of Cremona, Fibonacci in 1202 and Robert Recorde in 1551.
The history runs deep. As early as 1800 BCE, Babylonian scribes approximated the square root of 2 on the tablet YBC 7289 to the equivalent of six decimal places, and tablets from Larsa list square and cube roots of whole numbers. The Pythagorean Hippasus most likely first proved √2 irrational, and Plato's Theaetetus credits Theodorus of Cyrene, around 400 BC, with further such proofs. In the first century AD Heron of Alexandria devised a repeating procedure for square roots, a special case of what is now called Newton's method.
Source: Nth root
