nth root

In mathematics, an nth root of a number x is a number r which, when raised to the power n, yields x:

where n is a positive integer, sometimes called the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an nth root is a root extraction.

For example, 3 is a square root of 9, since 32 = 9, and −3 is also a square root of 9, since (−3)2 = 9.

Any non-zero number considered as a complex number has n different complex nth roots, including the real ones (at most two). The nth root of 0 is zero for all positive integers n, since 0n = 0. In particular, if n is even and x is a positive real number, one of its nth roots is real and positive, one is negative, and the others (when n > 2) are non-real complex numbers; if n is even and x is a negative real number, none of the nth roots is real. If n is odd and x is real, one nth root is real and has the same sign as x, while the other (n – 1) roots are not real. Finally, if x is not real, then none of its nth roots are real.

When complex nth roots are considered, it is often useful to choose one of the roots, called principal root, as a principal value. The common choice is to choose the principal nth root of x as the nth root, with the greatest real part, and, when there are two (for x real and negative), the one with a positive imaginary part. This makes the nth root a function that is real and positive for x real and positive, and is continuous in the whole complex plane, except for values of x that are real and negative.

An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd[1] or a radical.[2] Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression, and if it contains no transcendental functions or transcendental numbers it is called an algebraic expression.

Roots can also be defined as special cases of exponentiation, where the exponent is a fraction:

Roots are used for determining the radius of convergence of a power series with the root test. The nth roots of 1 are called roots of unity and play a fundamental role in various areas of mathematics, such as number theory, theory of equations, and Fourier transform.

An archaic term for the operation of taking nth roots is radication.[3][4]

An nth root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x:

Every non-zero number x, real or complex, has n different complex number nth roots. (In the case x is real, this count includes any real nth roots.) The only complex root of 0 is 0.

The nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two nth powers) are irrational. For example,

A square root of a number x is a number r which, when squared, becomes x:

Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign:

Since the square of every real number is nonnegative, negative numbers do not have real square roots. However, for every negative real number there are two imaginary square roots. For example, the square roots of −25 are 5i and −5i, where i represents a number whose square is −1.

Subtleties can occur when taking the nth roots of negative or complex numbers. For instance:

A non-nested radical expression is said to be in simplified form if[6]

Next, there is a fraction under the radical sign, which we change as follows:

When there is a denominator involving surds it is always possible to find a factor to multiply both numerator and denominator by to simplify the expression.[7][8] For instance using the factorization of the sum of two cubes:

Simplifying radical expressions involving nested radicals can be quite difficult. It is not obvious for instance that:

The nth root of a number A can be computed with Newton's method, which starts with an initial guess x0 and then iterates using the recurrence relation

until the desired precision is reached. For computational efficiency, the recurrence relation is commonly rewritten

This allows to have only one exponentiation, and to compute once for all the first factor of each term.

For example, to find the fifth root of 34, we plug in n = 5, A = 34 and x0 = 2 (initial guess). The first 5 iterations are, approximately:
x0 = 2
x1 = 2.025
x2 = 2.02439 7...
x3 = 2.02439 7458...
x4 = 2.02439 74584 99885 04251 08172...

(All correct digits shown.)

x5 = 2.02439 74584 99885 04251 08172 45541 93741 91146 21701 07311 8...

The approximation x4 is accurate to 25 decimal places and x5 is good for 51.

Newton's method can be modified to produce various generalized continued fraction for the nth root. For example,

Digit-by-digit calculation of principal roots of decimal (base 10) numbers

Write the original number in decimal form. The numbers are written similar to the long division algorithm, and, as in long division, the root will be written on the line above. Now separate the digits into groups of digits equating to the root being taken, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the radicand. One digit of the root will appear above each group of digits of the original number.

Beginning with the left-most group of digits, do the following procedure for each group:

(Note: That formula shows b raised to the power of the result of the division, not b multiplied by the result of the division.)

The ancient Greek mathematicians knew how to use compass and straightedge to construct a length equal to the square root of a given length, when an auxiliary line of unit length is given. In 1837 Pierre Wantzel proved that an nth root of a given length cannot be constructed if n is not a power of 2.[9]

The two square roots of a complex number are always negatives of each other. For example, the square roots of −4 are 2i and −2i, and the square roots of i are

If we express a complex number in polar form, then the square root can be obtained by taking the square root of the radius and halving the angle:

A principal root of a complex number may be chosen in various ways, for example

which introduces a branch cut in the complex plane along the positive real axis with the condition 0 ≤ θ < 2π, or along the negative real axis with π < θ ≤ π.

Every complex number has n different nth roots in the complex plane. These are

where η is a single nth root, and 1, ωω2, ... ωn−1 are the nth roots of unity. For example, the four different fourth roots of 2 are

If n is even, a complex number's nth roots, of which there are an even number, come in additive inverse pairs, so that if a number r1 is one of the nth roots then r2 = –r1 is another. This is because raising the latter's coefficient –1 to the nth power for even n yields 1: that is, (–r1)n = (–1)n × r1n = r1n.

As with square roots, the formula above does not define a continuous function over the entire complex plane, but instead has a branch cut at points where θ / n is discontinuous.

It was once conjectured that all polynomial equations could be solved algebraically (that is, that all roots of a polynomial could be expressed in terms of a finite number of radicals and elementary operations). However, while this is true for third degree polynomials (cubics) and fourth degree polynomials (quartics), the Abel–Ruffini theorem (1824) shows that this is not true in general when the degree is 5 or greater. For example, the solutions of the equation