Nth root

In mathematics, an "n"th 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 th root is a root extraction.
For example, 3 is a square root of 9, since 3 = 9, and −3 is also a square root of 9, since (−3) = 9.
Any non-zero number considered as a complex number has different complex th roots, including the real ones (at most two). The th root of 0 is zero for all positive integers , since . In particular, if is even and is a positive real number, one of its th roots is real and positive, one is negative, and the others (when ) are non-real complex numbers; if is even and is a negative real number, none of the th roots is real. If is odd and is real, one th root is real and has the same sign as , while the other () roots are not real. Finally, if is not real, then none of its th roots are real.
Roots of real numbers are usually written using the radical symbol or "radix" formula_2
Subtleties can occur when taking the "n"th roots of negative or complex numbers. For instance:
Since the rule formula_5 strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.
Simplified form of a radical expression.
A non-nested radical expression is said to be in simplified form if
For example, to write the radical expression formula_6 in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:
Next, there is a fraction under the radical sign, which we change as follows:
Finally, we remove the radical from the denominator 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. 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 above can be derived through:
Let formula_13, with and coprime and positive integers. Then formula_14 is rational if and only if both formula_15 and formula_16 are integers, which means that both formula_17 and formula_18 are "n"th powers of some integer.
Infinite series.
The radical or root may be represented by the infinite series:
with formula_20. This expression can be derived from the binomial series.
Computing principal roots.
Using Newton's method.
The "n"th root of a number "A" can be computed with Newton's method. Start with an initial guess "x"0 and then iterate using the recurrence relation
until the desired precision is reached.
Depending on the application, it may be enough to use only the first Newton approximant:
For example, to find the fifth root of 34, note that 2 = 32 and thus take "x" = 2, "n" = 5 and "y" = 2 in the above formula. This yields
The error in the approximation is only about 0.03%.
Newton's method can be modified to produce a generalized continued fraction for the "n"th root which can be modified in various ways as described in that article. For example:
In the case of the fifth root of 34 above (after dividing out selected common factors):
Digit-by-digit calculation of principal roots of decimal (base 10) numbers.
Building on the digit-by-digit calculation of a square root, it can be seen that the formula used there, formula_27, or formula_28, follows a pattern involving Pascal's triangle. For the "n"th root of a number formula_29 is defined as the value of element formula_30 in row formula_31 of Pascal's Triangle such that formula_32, we can rewrite the expression as formula_33. For convenience, call the result of this expression formula_34. Using this more general expression, any positive principal root can be computed, digit-by-digit, as follows.
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:
Examples.
Find the square root of 152.2756.
 1 2. 3 4 
 \/ 01 52.27 56
 01 10·1·0·1 + 10·2·0·1 ≤ 1 < 10·1·0·2 + 10·2·0·2 x = 1
 01 y = 10·1·0·1 + 10·2·0·1 = 1 + 0 = 1
 00 52 10·1·1·2 + 10·2·1·2 ≤ 52 < 10·1·1·3 + 10·2·1·3 x = 2
 00 44 y = 10·1·1·2 + 10·2·1·2 = 4 + 40 = 44
 08 27 10·1·12·3 + 10·2·12·3 ≤ 827 < 10·1·12·4 + 10·2·12·4 x = 3
 07 29 y = 10·1·12·3 + 10·2·12·3 = 9 + 720 = 729
 98 56 10·1·123·4 + 10·2·123·4 ≤ 9856 < 10·1·123·5 + 10·2·123·5 x = 4
 98 56 y = 10·1·123·4 + 10·2·123·4 = 16 + 9840 = 9856
 00 00 Algorithm terminates: Answer is 12.34
Find the cube root of 4192 to the nearest hundredth.
 1 6. 1 2 4
 3 /
 \/ 004 192.000 000 000
 004 10·1·0·1 + 10·3·0·1 + 10·3·0·1 ≤ 4 < 10·1·0·2 + 10·3·0·2 + 10·3·0·2 x = 1
 001 y = 10·1·0·1 + 10·3·0·1 + 10·3·0·1 = 1 + 0 + 0 = 1
 003 192 10·1·1·6 + 10·3·1·6 + 10·3·1·6 ≤ 3192 < 10·1·1·7 + 10·3·1·7 + 10·3·1·7 x = 6
 003 096 y = 10·1·1·6 + 10·3·1·6 + 10·3·1·6 = 216 + 1,080 + 1,800 = 3,096
 096 000 10·1·16·1 + 10·3·16·1 + 10·3·16·1 ≤ 96000 < 10·1·16·2 + 10·3·16·2 + 10·3·16·2 x = 1
 077 281 y = 10·1·16·1 + 10·3·16·1 + 10·3·16·1 = 1 + 480 + 76,800 = 77,281
 018 719 000 10·1·161·2 + 10·3·161·2 + 10·3·161·2 ≤ 18719000 < 10·1·161·3 + 10·3·161·3 + 10·3·161·3 x = 2
 015 571 928 y = 10·1·161·2 + 10·3·161·2 + 10·3·161·2 = 8 + 19,320 + 15,552,600 = 15,571,928
 003 147 072 000 10·1·1612·4 + 10·3·1612·4 + 10·3·1612·4 ≤ 3147072000 < 10·1·1612·5 + 10·3·1612·5 + 10·3·1612·5 x = 4
 The desired precision is achieved:
 The cube root of 4192 is about 16.12
Logarithmic calculation.
The principal "n"th root of a positive number can be computed using logarithms. Starting from the equation that defines "r" as an "n"th root of "x", namely formula_43 with "x" positive and therefore its principal root "r" also positive, one takes logarithms of both sides (any base of the logarithm will do) to obtain
The root "r" is recovered from this by taking the antilog:
For the case in which "x" is negative and "n" is odd, there is one real root "r" which is also negative. This can be found by first multiplying both sides of the defining equation by −1 to obtain formula_46 then proceeding as before to find |"r"|, and using .
Geometric constructibility.
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 "n"th root of a given length cannot be constructed if "n" is not a power of 2.
Complex roots.
Every complex number other than 0 has "n" different "n"th roots.
Square roots.
The two square roots of a complex number are always negatives of each other. For example, the square roots of are and , and the square roots of 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 , or along the negative real axis with .
Using the first(last) branch cut the principal square root formula_50 maps formula_51 to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed in mathematical software like Matlab or Scilab.
Roots of unity.
The number 1 has "n" different "n"th roots in the complex plane, namely
where
These roots are evenly spaced around the unit circle in the complex plane, at angles which are multiples of formula_54. For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, formula_30, −1, and formula_56.
"n"th roots.
Every complex number has "n" different "n"th roots in the complex plane. These are
where "η" is a single "n"th root, and 1, "ω", "ω", ... "ω" are the "n"th roots of unity. For example, the four different fourth roots of 2 are
In polar form, a single "n"th root may be found by the formula
Here "r" is the magnitude (the modulus, also called the absolute value) of the number whose root is to be taken; if the number can be written as "a+bi" then formula_60. Also, formula_61 is the angle formed as one pivots on the origin counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the properties that formula_62 formula_63 and formula_64
Thus finding "n"th roots in the complex plane can be segmented into two steps. First, the magnitude of all the "n"th roots is the "n"th root of the magnitude of the original number. Second, the angle between the positive horizontal axis and a ray from the origin to one of the "n"th roots is formula_65, where formula_61 is the angle defined in the same way for the number whose root is being taken. Furthermore, all "n" of the "n"th roots are at equally spaced angles from each other.
If "n" is even, a complex number's "n"th roots, of which there are an even number, come in additive inverse pairs, so that if a number "r"1 is one of the "n"th roots then "r"2 = –"r"1 is another. This is because raising the latter's coefficient –1 to the "n"th power for even "n" yields 1: that is, (–"r"1) = (–1) × "r"1 = "r"1.
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.
Solving polynomials.
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
cannot be expressed in terms of radicals. ("cf." quintic equation)
Proof of irrationality for non-perfect "n"th power "x".
Assume that formula_68 is rational. That is, it can be reduced to a fraction formula_69, where and are integers without a common factor.
This means that formula_70.
Since "x" is an integer, formula_71and formula_72must share a common factor if formula_73. This means that if formula_73, formula_75 is not in simplest form. Thus "b" should equal 1.
Since formula_76 and formula_77, formula_78.
This means that formula_79 and thus, formula_80. This implies that formula_68 is an integer. Since "x" is not a perfect "n"th power, this is impossible. Thus formula_68 is irrational.