{"history":[],"article":{"id":"e8daee19-e7e0-42cc-9d31-16d2334634fa","title":"Nth root","slug":"nth-root","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), an **nth root** of a [number](https://bliptext.com/articles/number) x is a [number](https://bliptext.com/articles/number) r (the root) which, when [raised to the power](https://bliptext.com/articles/exponentiation) of the [positive integer](https://bliptext.com/articles/positive-integer) n, yields x:\nThe integer n is called the index or degree, and the number x of which the root is taken is the radicand. A root of degree 2 is called a [square root](https://bliptext.com/articles/square-root) and a root of degree 3, a [cube root](https://bliptext.com/articles/cube-root).\nRoots of higher degree are referred by using [ordinal numbers](https://bliptext.com/articles/ordinal-numeral), as in fourth root, twentieth root, etc.\nThe computation of an nth root is a **root extraction**.\nFor example,\n3\nis a square root of\n9\n, since\n32 = 9\n, and\n−3\nis also a square root of\n9\n, since\n(−3)2 = 9\n.\nThe nth root of x is written as \\sqrt[n]{x} using the [radical symbol](https://bliptext.com/articles/radical-symbol) or radix.\nThe square root is usually written without the n as just \\sqrt x. Taking the nth root of a number is the [inverse](https://bliptext.com/articles/inverse-function) operation of [exponentiation](https://bliptext.com/articles/exponentiation), and can be written as a [fractional](https://bliptext.com/articles/fraction-mathematics) exponent:\nFor a positive real number x, \\sqrt{x} denotes the positive square root of x and \\sqrt[n]{x} denotes the positive real nth root.\nA negative real number\n−x\nhas no real-valued square roots, but when x is treated as a complex number it has two [imaginary](https://bliptext.com/articles/imaginary-number) square roots, +i\\sqrt x and -i\\sqrt x, where i is the [imaginary](https://bliptext.com/articles/imaginary-number) unit.\nIn general, any non-zero [complex number](https://bliptext.com/articles/complex-number) has n distinct complex-valued nth roots, equally distributed around a complex circle of constant [absolute value](https://bliptext.com/articles/absolute-value).\n(The nth root of\n0\nis zero with [multiplicity](https://bliptext.com/articles/multiple-root) n, and this circle degenerates to a point.) Extracting the nth roots of a complex number x can thus be taken to be a [multivalued function](https://bliptext.com/articles/multivalued-[function](https://bliptext.com/articles/function-mathematics)). By convention the [principal value](https://bliptext.com/articles/principal-value) of this [function](https://bliptext.com/articles/function-mathematics), called the **principal root** and denoted \\sqrt[n]{x}, is taken to be the nth root with the greatest real part and in the special case when x is a negative real number, the one with a positive [imaginary part](https://bliptext.com/articles/imaginary-part). The **principal root** of a positive real number is thus also a positive real number. As a [function](https://bliptext.com/articles/function-mathematics), the **principal root** is [continuous](https://bliptext.com/articles/continuous-function) in the whole [complex plane](https://bliptext.com/articles/complex-plane), except along the negative real axis.\nAn unresolved root, especially one using the **radical** symbol, is sometimes referred to as a **surd** or a **radical**.\nAny 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](https://bliptext.com/articles/transcendental-functions) or [transcendental numbers](https://bliptext.com/articles/transcendental-numbers) it is called an [algebraic expression](https://bliptext.com/articles/algebraic-expression).\nRoots are used for determining the [radius of convergence](https://bliptext.com/articles/radius-of-convergence) of a [power series](https://bliptext.com/articles/power-series) with the [root test](https://bliptext.com/articles/root-test).\nThe nth roots of 1 are called [roots of unity](https://bliptext.com/articles/roots-of-unity) and play a fundamental role in various areas of mathematics, such as [number theory](https://bliptext.com/articles/number-theory), [theory of equations](https://bliptext.com/articles/theory-of-equations), and [Fourier transform](https://bliptext.com/articles/fourier-transform).\n\n# History\nAn archaic term for the operation of taking nth roots is radication.\n\n# Definition and notation\nAn 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:\nr^n = x.\nEvery positive [real number](https://bliptext.com/articles/real-number) x has a single positive nth root, called the principal nth root, which is written \\sqrt[n]{x}.\nFor n equal to 2 this is called the principal square root and the n is omitted.\nThe nth root can also be represented using [exponentiation](https://bliptext.com/articles/exponentiation) as x1/n.\nFor even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nth root.\nFor odd values of n, every negative number x has a real negative nth root.\nFor example, −2 has a real 5th root, but −2 does not have any real 6th roots.\nEvery non-zero number x, real or [complex](https://bliptext.com/articles/complex-number), has n different [complex](https://bliptext.com/articles/complex-number) number nth roots.\n(In the case x is real, this count includes any real nth roots.) The only complex root of 0 is 0.\nThe nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two nth powers) are [irrational](https://bliptext.com/articles/irrational-number).\nFor example,\nAll nth roots of rational numbers are [algebraic numbers](https://bliptext.com/articles/algebraic-number), and all nth roots of integers are [algebraic integers](https://bliptext.com/articles/algebraic-integer).\nThe term \"surd\" traces back to [Al-Khwarizmi](https://bliptext.com/articles/al-khwarizmi) (c. 825), who referred to rational and irrational numbers as audible and inaudible, respectively.\nThis later led to the Arabic word أصم (asamm, meaning \"deaf\" or \"dumb\") for irrational number being translated into Latin as surdus (meaning \"deaf\" or \"mute\").\n[Gerard of Cremona](https://bliptext.com/articles/gerard-of-cremona) (c. 1150), [Fibonacci](https://bliptext.com/articles/fibonacci) (1202), and then [Robert Recorde](https://bliptext.com/articles/robert-recorde) (1551) all used the term to refer to unresolved irrational roots, that is, expressions of the form \\sqrt[n]{r}, in which n and r are integer numerals and the whole expression denotes an irrational number.\nIrrational numbers of the form where a is rational, are called pure quadratic surds; irrational numbers of the form , where a and b are rational, are called [mixed quadratic surds](https://bliptext.com/articles/quadratic-irrational-number).\n# [The four 4th roots of −1,\nnone of which are real | upload.wikimedia.org/wikipedia/commons/a/aa/NegativeOne4Root.svg]\n# [The three 3rd roots of −1,\none of which is a negative real | upload.wikimedia.org/wikipedia/commons/9/91/NegativeOne3Root.svg]\n\n## Square roots\nA **square root** of a number x is a number r which, when [squared](https://bliptext.com/articles/square-algebra), becomes x:\nr^2 = x.\nEvery positive real number has two square roots, one positive and one negative.\nFor example, the two square roots of 25 are 5 and −5.\nThe positive square root is also known as the **principal square root**, and is denoted with a radical sign:\nSince the square of every real number is nonnegative, negative numbers do not have real square roots.\nHowever, for every negative real number there are two [imaginary](https://bliptext.com/articles/imaginary-number) square roots.\nFor example, the square roots of −25 are 5[i](https://bliptext.com/articles/imaginary-unit) and −5[i](https://bliptext.com/articles/imaginary-unit), where [i](https://bliptext.com/articles/imaginary-unit) represents a number whose square [i](https://bliptext.com/articles/imaginary-unit)s\n−1\n.\n# [The graph. | upload.wikimedia.org/wikipedia/commons/2/2e/Square-root///function.svg]\n\n## Cube roots\nA **[cube](https://bliptext.com/articles/cube-algebra) root** of a number x is a number r whose [cube](https://bliptext.com/articles/cube-algebra) is x:\nr^3 = x.\nEvery real number x has exactly one real cube root, written \\sqrt[3]{x}.\nFor example,\nEvery real number has two additional [complex](https://bliptext.com/articles/complex-number) cube roots.\n# [The graph. | upload.wikimedia.org/wikipedia/commons/e/e2/cube-root///function.svg]\n\n# Identities and properties\nExpressing the degree of an nth root in its exponent form, as in x^{1/n}, makes it easier to manipulate powers and roots.\nIf a is a [non-negative real number](https://bliptext.com/articles/non-negative-number),\nEvery non-negative number has exactly one non-negative real nth root, and so the rules for operations with surds involving non-negative radicands a and b are straightforward within the real numbers:\nSubtleties can occur when taking the nth roots of negative or [complex numbers](https://bliptext.com/articles/complex-number).\nFor instance:\nbut, rather,\nSince the rule strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.\n\n# Simplified form of a radical expression\nA [non-nested radical expression](https://bliptext.com/articles/nested-radical) is said to be in **simplified form** if no factor of the radicand can be written as a power greater than or equal to the index; there are no fractions inside the radical sign; and there are no radicals in the denominator.\nFor example, to write the radical expression in simplified form, we can proceed as follows.\nFirst, look for a perfect square under the square root sign and remove it:\nNext, there is a fraction under the radical sign, which we change as follows:\nFinally, we remove the radical from the denominator as follows:\nWhen 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.\nFor instance using the [factorization of the sum of two cubes](https://bliptext.com/articles/factorization):\nSimplifying radical expressions involving [nested radicals](https://bliptext.com/articles/nested-radical) can be quite difficult.\nIn particular, denesting is not always possible, and when possible, it may involve advanced [Galois theory](https://bliptext.com/articles/galois-theory).\nMoreover, when complete denesting is impossible, there is no general [canonical form](https://bliptext.com/articles/canonical-form) such that the equality of two numbers can be tested by simply looking at their canonical expressions.\nFor example, it is not obvious that\nThe above can be derived through:\nLet r=p/q, with p and q coprime and positive integers.\nThen is rational if and only if both \\sqrt[n]{p} and \\sqrt[n]{q} are integers, which means that both p and q are nth powers of some integer.\n\n# Infinite series\nThe radical or root may be represented by the [infinite series](https://bliptext.com/articles/infinite-series):\nwith |x|<1.\nThis expression can be derived from the [binomial series](https://bliptext.com/articles/binomial-series).\n\n# Computing principal roots\n\n## Using Newton's method\nThe nth root of a number\nA\ncan be computed with [Newton's method](https://bliptext.com/articles/newton-s-method), which starts with an initial guess\nx0\nand then iterates using the [recurrence relation](https://bliptext.com/articles/recurrence-relation)\nuntil the desired precision is reached.\nFor computational efficiency, the recurrence relation is commonly rewritten\nThis allows to have only one [exponentiation](https://bliptext.com/articles/exponentiation), and to compute once for all the first factor of each term.\nFor example, to find the fifth root of 34, we plug in\nn = 5, A = 34\nand\nx0 = 2\n(initial guess).\nThe first 5 iterations are, approximately:\n(All correct digits shown.)\nThe approximation\nx4\nis accurate to 25 decimal places and\nx5\nis good for 51.\nNewton's method can be modified to produce various [generalized continued fractions](https://bliptext.com/articles/generalized-continued-fraction) for the nth root.\nFor example,\n\n## Digit-by-digit calculation of principal roots of decimal (base 10) numbers\nBuilding on the [digit-by-digit calculation of a square root](https://bliptext.com/articles/methods-of-computing-square-roots), it can be seen that the formula used there,, or , follows a pattern involving Pascal's triangle.\nFor the nth root of a number P(n,i) is defined as the value of element i in row n of Pascal's Triangle such that P(4,1) = 4, we can rewrite the expression as.\nFor convenience, call the result of this expression y. Using this more general expression, any positive principal root can be computed, digit-by-digit, as follows.\nWrite the original number in decimal form.\nThe numbers are written similar to the [long division](https://bliptext.com/articles/long-division) algorithm, and, as in [long division](https://bliptext.com/articles/long-division), the root will be written on the line above.\nNow separate the digits into groups of digits equating to the root being taken, starting from the decimal point and going both left and right.\nThe decimal point of the root will be above the decimal point of the radicand.\nOne digit of the root will appear above each group of digits of the original number.\nBeginning with the left-most group of digits, do the following procedure for each group:\n# [Pascal's triangle showing P(4,1) = 4. | upload.wikimedia.org/wikipedia/commons/c/c6/PascalForDecimalRoots.svg]\n\n### Examples\n**Find the square root of 152.2756.**\n1 2. 3 4 / \\/ 01 52.27 56 (Results) (Explanations)\n0****1**** x = ****1**** ****1****00·****1****·00·****1****2 + ****1****0****1****·2·0****1****·****1********1**** ≤ ****1**** < ****1****00·****1****·00·22 + ****1****0****1****·2·0****1****·2****1****\n0**1** y = **1** y = **1**00·**1**·00·**1**2 + **1**0**1**·2·0**1**·**1****1** = **1** + 0 = **1**\n00 5****2**** x = ****2**** 100·1·10·****2********2**** + 101·****2****·11·****2****1 ≤ 5****2**** < 100·1·10·3****2**** + 101·****2****·11·31\n00 **44** y = **44** y = 100·1·10·22 + 101·2·11·21 = 4 + 40 = **44**\n08 27 x = ****3**** 100·1·120·****3****2 + 101·2·121·****3****1 ≤ 827 < 100·1·120·42 + 101·2·121·41\n07 29 y = **729** y = 100·1·120·32 + 101·2·121·31 = 9 + 720 = **729**\n98 56 x = ****4**** 100·1·1230·****4****2 + 101·2·1231·****4****1 ≤ 9856 < 100·1·1230·52 + 101·2·1231·51\n98 56 y = **9856** y = 100·1·1230·42 + 101·2·1231·41 = 16 + 9840 = **9856**\n00 00\nAlgorithm terminates: Answer is 12.34\n**Find the cube root of 4192 truncated to the nearest thousandth.**\n1 6. 1 2 4 3 / \\/ 004 192.000 000 000 (Results) (Explanations)\n004 x = ******1****** ******1******00·******1******·00·******1******3 + ******1******0******1******·3·0******1******·******1******2 + ******1******02·3·02·******1************1****** ≤ 4 < ******1******00·******1******·00·23 + ******1******0******1******·3·0******1******·22 + ******1******02·3·02·2******1******\n00**1** y = **1** y = **1**00·**1**·00·**1**3 + **1**0**1**·3·0**1**·**1**2 + **1**02·3·02·**1****1** = **1** + 0 + 0 = **1**\n003 192 x = ******6****** 100·1·10·******6******3 + 101·3·11·******6******2 + 102·3·12·******6******1 ≤ 3192 < 100·1·10·73 + 101·3·11·72 + 102·3·12·71\n003 096 y = 3096 y = 100·1·10·63 + 101·3·11·62 + 102·3·12·61 = 216 + 1,080 + 1,800 = **3,096**\n096 000 x = ******1****** ******1******00·******1******·******1******60·******1******3 + ******1******0******1******·3·******1******6******1******·******1******2 + ******1******02·3·******1******62·******1************1****** ≤ 96000 < ******1******00·******1******·******1******60·23 + ******1******0******1******·3·******1******6******1******·22 + ******1******02·3·******1******62·2******1******\n077 281 y = 77281 y = 100·1·160·13 + 101·3·161·12 + 102·3·162·11 = 1 + 480 + 76,800 = **77,281**\n018 719 000 x = ******2****** 100·1·1610·******2******3 + 101·3·1611·******2************2****** + 10******2******·3·161******2******·******2******1 ≤ 18719000 < 100·1·1610·33 + 101·3·1611·3******2****** + 10******2******·3·161******2******·31\n015 571 928 y = 15571928 y = 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 = 8 + 19,320 + 15,552,600 = **15,571,928**\n003 1******4******7 072 000 x = ******4****** 100·1·16120·******4******3 + 101·3·16121·******4******2 + 102·3·16122·******4******1 ≤ 31******4******7072000 < 100·1·16120·53 + 101·3·16121·52 + 102·3·16122·51\nThe desired precision is achieved.\nThe cube root of 4192 is 16.124...\n\n## Logarithmic calculation\nThe principal nth root of a positive number can be computed using [logarithms](https://bliptext.com/articles/logarithm).\nStarting from the equation that defines r as an nth root of x, namely r^n=x, with x positive and therefore its principal root r also positive, one takes logarithms of both sides (any [base of the logarithm](https://bliptext.com/articles/logarithm) will do) to obtain\nThe root r is recovered from this by taking the [antilog](https://bliptext.com/articles/antilog):\n(Note: That formula shows b raised to the power of the result of the division, not b multiplied by the result of the division.)\nFor the case in which x is negative and n is odd, there is one real root r which is also negative.\nThis can be found by first multiplying both sides of the defining equation by −1 to obtain then proceeding as before to find |r|, and using r.\n\n# Geometric constructibility\nThe [ancient Greek mathematicians](https://bliptext.com/articles/ancient-greek-mathematicians) knew how to [use compass and straightedge](https://bliptext.com/articles/compass-and-straightedge-construction) to construct a length equal to the square root of a given length, when an auxiliary line of unit length is given.\nIn 1837 [Pierre Wantzel](https://bliptext.com/articles/pierre-wantzel) proved that an nth root of a given length cannot be constructed if n is not a power of 2.\n\n# Complex roots\nEvery [complex number](https://bliptext.com/articles/complex-number) other than 0 has n different nth roots.\n\n## Square roots\nThe two square roots of a complex number are always negatives of each other.\nFor example, the square roots of\n−4\nare\n2i\nand\n−2i\n, and the square roots of\ni\nare\nIf we express a complex number in [polar form](https://bliptext.com/articles/polar-form), then the square root can be obtained by taking the square root of the radius and halving the angle:\nA principal root of a complex number may be chosen in various ways, for example\nwhich introduces a [branch cut](https://bliptext.com/articles/branch-cut) in the [complex plane](https://bliptext.com/articles/complex-plane) along the [positive real axis](https://bliptext.com/articles/positive-real-axis) with the condition\n0 ≤ θ < 2π\n, or along the negative real axis with\n−π < θ ≤ π\n.\nUsing the first(last) branch cut the principal square root maps to the half plane with non-negative imaginary(real) part.\nThe last branch cut is presupposed in mathematical software like [Matlab](https://bliptext.com/articles/matlab) or [Scilab](https://bliptext.com/articles/scilab).\n# [The square roots of i | upload.wikimedia.org/wikipedia/commons/3/32/Imaginary2Root.svg]\n\n## Roots of unity\nThe number 1 has n different nth roots in the complex plane, namely\nwhere\nThese roots are evenly spaced around the [unit circle](https://bliptext.com/articles/unit-circle) in the complex plane, at angles which are multiples of 2\\pi/n.\nFor example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, i, −1, and -i.\n# [The three 3rd roots of 1 | upload.wikimedia.org/wikipedia/commons/3/39/3rd///roots///of///unity.svg]\n\n## nth roots\nEvery complex number has n different nth roots in the complex plane.\nThese are\nwhere η is a single nth root, and 1, ω, ω2, ... ωn−1 are the nth roots of unity.\nFor example, the four different fourth roots of 2 are\nIn [polar form](https://bliptext.com/articles/polar-form), a single nth root may be found by the formula\nHere r is the magnitude (the modulus, also called the [absolute value](https://bliptext.com/articles/absolute-value)) of the number whose root is to be taken; if the number can be written as a+bi then.\nAlso, \\theta 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 and\nThus finding nth roots in the complex plane can be segmented into two steps.\nFirst, the magnitude of all the nth roots is the nth root of the magnitude of the original number.\nSecond, the angle between the positive horizontal axis and a ray from the origin to one of the nth roots is \\theta / n, where \\theta is the angle defined in the same way for the number whose root is being taken.\nFurthermore, all n of the nth roots are at equally spaced angles from each other.\nIf n is even, a complex number's nth roots, of which there are an even number, come in [additive inverse](https://bliptext.com/articles/additive-inverse) pairs, so that if a number r1 is one of the nth roots then r2 = –r1 is another.\nThis 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.\nAs with square roots, the formula above does not define a [continuous function](https://bliptext.com/articles/continuous-function) over the entire complex plane, but instead has a [branch cut](https://bliptext.com/articles/branch-cut) at points where θ / n is discontinuous.\n\n# Solving polynomials\nIt was once [conjectured](https://bliptext.com/articles/conjecture) that all [polynomial equations](https://bliptext.com/articles/[polynomial](https://bliptext.com/articles/polynomial)-equation) could be [solved algebraically](https://bliptext.com/articles/algebraic-solution) (that is, that all roots of a [polynomial](https://bliptext.com/articles/polynomial) could be expressed in terms of a finite number of radicals and [elementary operations](https://bliptext.com/articles/elementary-arithmetic)).\nHowever, while this is true for third degree polynomials ([cubics](https://bliptext.com/articles/cubic-function)) and fourth degree polynomials ([quartics](https://bliptext.com/articles/quartic-function)), the [Abel–Ruffini theorem](https://bliptext.com/articles/abel-ruffini-theorem) (1824) shows that this is not true in general when the degree is 5 or greater.\nFor example, the solutions of the equation\nx^5 = x + 1\ncannot be expressed in terms of radicals.\n(cf. [quintic equation](https://bliptext.com/articles/quintic-equation))\n\n# Proof of irrationality for non-perfect nth power x\nAssume that \\sqrt[n]{x} is rational.\nThat is, it can be reduced to a fraction \\frac{a}{b}, where a and b are integers without a common factor.\nThis means that.\nSince x is an integer, a^nand b^nmust share a common factor if b \\neq 1.\nThis means that if b \\neq 1, is not in simplest form.\nThus b should equal 1.\nSince 1^n = 1 and,.\nThis means that x = a^n and thus,.\nThis implies that \\sqrt[n]{x} is an integer.\nSince x is not a perfect nth power, this is impossible.\nThus \\sqrt[n]{x} is irrational.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:49:12.458Z","updated_at":"2025-01-25T18:49:12.458Z","search_vector":"'nth':1 'root':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), an **nth root** of a [number](https://bliptext.com/articles/number) x is a [number](https://bliptext.com/articles/number) r (the root) which, when [raised to the power](https://bliptext.com/articles/exponentiation) of the [positive integer](https://bliptext.com/articles/positive-integer) n, yields x:\nThe integer n is called the index or degree, and the number x of which the root is taken is the radicand. A root of degree 2 is called a [square root](https://bliptext.com/articles/square-root) and a root of degree 3, a [cube root](https://bliptext.com/articles/cube-root).\nRoots of higher degree are referred by using [ordinal numbers](https://bliptext.com/articles/ordinal-numeral), as in fourth root, twentieth root, etc.\nThe computation of an nth root is a **root extraction**.\nFor example,\n3\nis a square root of\n9\n, since\n32 = 9\n, and\n−3\nis also a square root of\n9\n, since\n(−3)2 = 9\n.\nThe nth root of x is written as \\sqrt[n]{x} using the [radical symbol](https://bliptext.com/articles/radical-symbol) or radix.\nThe square root is usually written without the n as just \\sqrt x. Taking the nth root of a number is the [inverse](https://bliptext.com/articles/inverse-function) operation of [exponentiation](https://bliptext.com/articles/exponentiation), and can be written as a [fractional](https://bliptext.com/articles/fraction-mathematics) exponent:\nFor a positive real number x, \\sqrt{x} denotes the positive square root of x and \\sqrt[n]{x} denotes the positive real nth root.\nA negative real number\n−x\nhas no real-valued square roots, but when x is treated as a complex number it has two [imaginary](https://bliptext.com/articles/imaginary-number) square roots, +i\\sqrt x and -i\\sqrt x, where i is the [imaginary](https://bliptext.com/articles/imaginary-number) unit.\nIn general, any non-zero [complex number](https://bliptext.com/articles/complex-number) has n distinct complex-valued nth roots, equally distributed around a complex circle of constant [absolute value](https://bliptext.com/articles/absolute-value).\n(The nth root of\n0\nis zero with [multiplicity](https://bliptext.com/articles/multiple-root) n, and this circle degenerates to a point.) Extracting the nth roots of a complex number x can thus be taken to be a [multivalued [function](https://bliptext.com/articles/function-mathematics)](https://bliptext.com/articles/multivalued-[function](https://bliptext.com/articles/function-mathematics)). By convention the [principal value](https://bliptext.com/articles/principal-value) of this [function](https://bliptext.com/articles/function-mathematics), called the **principal root** and denoted \\sqrt[n]{x}, is taken to be the nth root with the greatest real part and in the special case when x is a negative real number, the one with a positive [imaginary part](https://bliptext.com/articles/imaginary-part). The **principal root** of a positive real number is thus also a positive real number. As a [function](https://bliptext.com/articles/function-mathematics), the **principal root** is [continuous](https://bliptext.com/articles/continuous-function) in the whole [complex plane](https://bliptext.com/articles/complex-plane), except along the negative real axis.\nAn unresolved root, especially one using the **radical** symbol, is sometimes referred to as a **surd** or a **radical**.\nAny 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](https://bliptext.com/articles/transcendental-functions) or [transcendental numbers](https://bliptext.com/articles/transcendental-numbers) it is called an [algebraic expression](https://bliptext.com/articles/algebraic-expression).\nRoots are used for determining the [radius of convergence](https://bliptext.com/articles/radius-of-convergence) of a [power series](https://bliptext.com/articles/power-series) with the [root test](https://bliptext.com/articles/root-test).\nThe nth roots of 1 are called [roots of unity](https://bliptext.com/articles/roots-of-unity) and play a fundamental role in various areas of mathematics, such as [number theory](https://bliptext.com/articles/number-theory), [theory of equations](https://bliptext.com/articles/theory-of-equations), and [Fourier transform](https://bliptext.com/articles/fourier-transform).\n\n# History\nAn archaic term for the operation of taking nth roots is radication.\n\n# Definition and notation\nAn 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:\nr^n = x.\nEvery positive [real number](https://bliptext.com/articles/real-number) x has a single positive nth root, called the principal nth root, which is written \\sqrt[n]{x}.\nFor n equal to 2 this is called the principal square root and the n is omitted.\nThe nth root can also be represented using [exponentiation](https://bliptext.com/articles/exponentiation) as x1/n.\nFor even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nth root.\nFor odd values of n, every negative number x has a real negative nth root.\nFor example, −2 has a real 5th root, but −2 does not have any real 6th roots.\nEvery non-zero number x, real or [complex](https://bliptext.com/articles/complex-number), has n different [complex](https://bliptext.com/articles/complex-number) number nth roots.\n(In the case x is real, this count includes any real nth roots.) The only complex root of 0 is 0.\nThe nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two nth powers) are [irrational](https://bliptext.com/articles/irrational-number).\nFor example,\nAll nth roots of rational numbers are [algebraic numbers](https://bliptext.com/articles/algebraic-number), and all nth roots of integers are [algebraic integers](https://bliptext.com/articles/algebraic-integer).\nThe term \"surd\" traces back to [Al-Khwarizmi](https://bliptext.com/articles/al-khwarizmi) (c. 825), who referred to rational and irrational numbers as audible and inaudible, respectively.\nThis later led to the Arabic word أصم (asamm, meaning \"deaf\" or \"dumb\") for irrational number being translated into Latin as surdus (meaning \"deaf\" or \"mute\").\n[Gerard of Cremona](https://bliptext.com/articles/gerard-of-cremona) (c. 1150), [Fibonacci](https://bliptext.com/articles/fibonacci) (1202), and then [Robert Recorde](https://bliptext.com/articles/robert-recorde) (1551) all used the term to refer to unresolved irrational roots, that is, expressions of the form \\sqrt[n]{r}, in which n and r are integer numerals and the whole expression denotes an irrational number.\nIrrational numbers of the form where a is rational, are called pure quadratic surds; irrational numbers of the form , where a and b are rational, are called [mixed quadratic surds](https://bliptext.com/articles/quadratic-irrational-number).\n# [The four 4th roots of −1,\nnone of which are real | upload.wikimedia.org/wikipedia/commons/a/aa/NegativeOne4Root.svg]\n# [The three 3rd roots of −1,\none of which is a negative real | upload.wikimedia.org/wikipedia/commons/9/91/NegativeOne3Root.svg]\n\n## Square roots\nA **square root** of a number x is a number r which, when [squared](https://bliptext.com/articles/square-algebra), becomes x:\nr^2 = x.\nEvery positive real number has two square roots, one positive and one negative.\nFor example, the two square roots of 25 are 5 and −5.\nThe positive square root is also known as the **principal square root**, and is denoted with a radical sign:\nSince the square of every real number is nonnegative, negative numbers do not have real square roots.\nHowever, for every negative real number there are two [imaginary](https://bliptext.com/articles/imaginary-number) square roots.\nFor example, the square roots of −25 are 5[i](https://bliptext.com/articles/imaginary-unit) and −5[i](https://bliptext.com/articles/imaginary-unit), where [i](https://bliptext.com/articles/imaginary-unit) represents a number whose square [i](https://bliptext.com/articles/imaginary-unit)s\n−1\n.\n# [The graph. | upload.wikimedia.org/wikipedia/commons/2/2e/Square-root///function.svg]\n\n## Cube roots\nA **[cube](https://bliptext.com/articles/cube-algebra) root** of a number x is a number r whose [cube](https://bliptext.com/articles/cube-algebra) is x:\nr^3 = x.\nEvery real number x has exactly one real cube root, written \\sqrt[3]{x}.\nFor example,\nEvery real number has two additional [complex](https://bliptext.com/articles/complex-number) cube roots.\n# [The graph. | upload.wikimedia.org/wikipedia/commons/e/e2/cube-root///function.svg]\n\n# Identities and properties\nExpressing the degree of an nth root in its exponent form, as in x^{1/n}, makes it easier to manipulate powers and roots.\nIf a is a [non-negative real number](https://bliptext.com/articles/non-negative-number),\nEvery non-negative number has exactly one non-negative real nth root, and so the rules for operations with surds involving non-negative radicands a and b are straightforward within the real numbers:\nSubtleties can occur when taking the nth roots of negative or [complex numbers](https://bliptext.com/articles/complex-number).\nFor instance:\nbut, rather,\nSince the rule strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.\n\n# Simplified form of a radical expression\nA [non-nested radical expression](https://bliptext.com/articles/nested-radical) is said to be in **simplified form** if no factor of the radicand can be written as a power greater than or equal to the index; there are no fractions inside the radical sign; and there are no radicals in the denominator.\nFor example, to write the radical expression in simplified form, we can proceed as follows.\nFirst, look for a perfect square under the square root sign and remove it:\nNext, there is a fraction under the radical sign, which we change as follows:\nFinally, we remove the radical from the denominator as follows:\nWhen 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.\nFor instance using the [factorization of the sum of two cubes](https://bliptext.com/articles/factorization):\nSimplifying radical expressions involving [nested radicals](https://bliptext.com/articles/nested-radical) can be quite difficult.\nIn particular, denesting is not always possible, and when possible, it may involve advanced [Galois theory](https://bliptext.com/articles/galois-theory).\nMoreover, when complete denesting is impossible, there is no general [canonical form](https://bliptext.com/articles/canonical-form) such that the equality of two numbers can be tested by simply looking at their canonical expressions.\nFor example, it is not obvious that\nThe above can be derived through:\nLet r=p/q, with p and q coprime and positive integers.\nThen is rational if and only if both \\sqrt[n]{p} and \\sqrt[n]{q} are integers, which means that both p and q are nth powers of some integer.\n\n# Infinite series\nThe radical or root may be represented by the [infinite series](https://bliptext.com/articles/infinite-series):\nwith |x|<1.\nThis expression can be derived from the [binomial series](https://bliptext.com/articles/binomial-series).\n\n# Computing principal roots\n\n## Using Newton's method\nThe nth root of a number\nA\ncan be computed with [Newton's method](https://bliptext.com/articles/newton-s-method), which starts with an initial guess\nx0\nand then iterates using the [recurrence relation](https://bliptext.com/articles/recurrence-relation)\nuntil the desired precision is reached.\nFor computational efficiency, the recurrence relation is commonly rewritten\nThis allows to have only one [exponentiation](https://bliptext.com/articles/exponentiation), and to compute once for all the first factor of each term.\nFor example, to find the fifth root of 34, we plug in\nn = 5, A = 34\nand\nx0 = 2\n(initial guess).\nThe first 5 iterations are, approximately:\n(All correct digits shown.)\nThe approximation\nx4\nis accurate to 25 decimal places and\nx5\nis good for 51.\nNewton's method can be modified to produce various [generalized continued fractions](https://bliptext.com/articles/generalized-continued-fraction) for the nth root.\nFor example,\n\n## Digit-by-digit calculation of principal roots of decimal (base 10) numbers\nBuilding on the [digit-by-digit calculation of a square root](https://bliptext.com/articles/methods-of-computing-square-roots), it can be seen that the formula used there,, or , follows a pattern involving Pascal's triangle.\nFor the nth root of a number P(n,i) is defined as the value of element i in row n of Pascal's Triangle such that P(4,1) = 4, we can rewrite the expression as.\nFor convenience, call the result of this expression y. Using this more general expression, any positive principal root can be computed, digit-by-digit, as follows.\nWrite the original number in decimal form.\nThe numbers are written similar to the [long division](https://bliptext.com/articles/long-division) algorithm, and, as in [long division](https://bliptext.com/articles/long-division), the root will be written on the line above.\nNow separate the digits into groups of digits equating to the root being taken, starting from the decimal point and going both left and right.\nThe decimal point of the root will be above the decimal point of the radicand.\nOne digit of the root will appear above each group of digits of the original number.\nBeginning with the left-most group of digits, do the following procedure for each group:\n# [Pascal's triangle showing P(4,1) = 4. | upload.wikimedia.org/wikipedia/commons/c/c6/PascalForDecimalRoots.svg]\n\n### Examples\n**Find the square root of 152.2756.**\n1 2. 3 4 / \\/ 01 52.27 56 (Results) (Explanations)\n0****1**** x = ****1**** ****1****00·****1****·00·****1****2 + ****1****0****1****·2·0****1****·****1********1**** ≤ ****1**** < ****1****00·****1****·00·22 + ****1****0****1****·2·0****1****·2****1****\n0**1** y = **1** y = **1**00·**1**·00·**1**2 + **1**0**1**·2·0**1**·**1****1** = **1** + 0 = **1**\n00 5****2**** x = ****2**** 100·1·10·****2********2**** + 101·****2****·11·****2****1 ≤ 5****2**** < 100·1·10·3****2**** + 101·****2****·11·31\n00 **44** y = **44** y = 100·1·10·22 + 101·2·11·21 = 4 + 40 = **44**\n08 27 x = ****3**** 100·1·120·****3****2 + 101·2·121·****3****1 ≤ 827 < 100·1·120·42 + 101·2·121·41\n07 29 y = **729** y = 100·1·120·32 + 101·2·121·31 = 9 + 720 = **729**\n98 56 x = ****4**** 100·1·1230·****4****2 + 101·2·1231·****4****1 ≤ 9856 < 100·1·1230·52 + 101·2·1231·51\n98 56 y = **9856** y = 100·1·1230·42 + 101·2·1231·41 = 16 + 9840 = **9856**\n00 00\nAlgorithm terminates: Answer is 12.34\n**Find the cube root of 4192 truncated to the nearest thousandth.**\n1 6. 1 2 4 3 / \\/ 004 192.000 000 000 (Results) (Explanations)\n004 x = ******1****** ******1******00·******1******·00·******1******3 + ******1******0******1******·3·0******1******·******1******2 + ******1******02·3·02·******1************1****** ≤ 4 < ******1******00·******1******·00·23 + ******1******0******1******·3·0******1******·22 + ******1******02·3·02·2******1******\n00**1** y = **1** y = **1**00·**1**·00·**1**3 + **1**0**1**·3·0**1**·**1**2 + **1**02·3·02·**1****1** = **1** + 0 + 0 = **1**\n003 192 x = ******6****** 100·1·10·******6******3 + 101·3·11·******6******2 + 102·3·12·******6******1 ≤ 3192 < 100·1·10·73 + 101·3·11·72 + 102·3·12·71\n003 096 y = 3096 y = 100·1·10·63 + 101·3·11·62 + 102·3·12·61 = 216 + 1,080 + 1,800 = **3,096**\n096 000 x = ******1****** ******1******00·******1******·******1******60·******1******3 + ******1******0******1******·3·******1******6******1******·******1******2 + ******1******02·3·******1******62·******1************1****** ≤ 96000 < ******1******00·******1******·******1******60·23 + ******1******0******1******·3·******1******6******1******·22 + ******1******02·3·******1******62·2******1******\n077 281 y = 77281 y = 100·1·160·13 + 101·3·161·12 + 102·3·162·11 = 1 + 480 + 76,800 = **77,281**\n018 719 000 x = ******2****** 100·1·1610·******2******3 + 101·3·1611·******2************2****** + 10******2******·3·161******2******·******2******1 ≤ 18719000 < 100·1·1610·33 + 101·3·1611·3******2****** + 10******2******·3·161******2******·31\n015 571 928 y = 15571928 y = 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 = 8 + 19,320 + 15,552,600 = **15,571,928**\n003 1******4******7 072 000 x = ******4****** 100·1·16120·******4******3 + 101·3·16121·******4******2 + 102·3·16122·******4******1 ≤ 31******4******7072000 < 100·1·16120·53 + 101·3·16121·52 + 102·3·16122·51\nThe desired precision is achieved.\nThe cube root of 4192 is 16.124...\n\n## Logarithmic calculation\nThe principal nth root of a positive number can be computed using [logarithms](https://bliptext.com/articles/logarithm).\nStarting from the equation that defines r as an nth root of x, namely r^n=x, with x positive and therefore its principal root r also positive, one takes logarithms of both sides (any [base of the logarithm](https://bliptext.com/articles/logarithm) will do) to obtain\nThe root r is recovered from this by taking the [antilog](https://bliptext.com/articles/antilog):\n(Note: That formula shows b raised to the power of the result of the division, not b multiplied by the result of the division.)\nFor the case in which x is negative and n is odd, there is one real root r which is also negative.\nThis can be found by first multiplying both sides of the defining equation by −1 to obtain then proceeding as before to find |r|, and using r.\n\n# Geometric constructibility\nThe [ancient Greek mathematicians](https://bliptext.com/articles/ancient-greek-mathematicians) knew how to [use compass and straightedge](https://bliptext.com/articles/compass-and-straightedge-construction) to construct a length equal to the square root of a given length, when an auxiliary line of unit length is given.\nIn 1837 [Pierre Wantzel](https://bliptext.com/articles/pierre-wantzel) proved that an nth root of a given length cannot be constructed if n is not a power of 2.\n\n# Complex roots\nEvery [complex number](https://bliptext.com/articles/complex-number) other than 0 has n different nth roots.\n\n## Square roots\nThe two square roots of a complex number are always negatives of each other.\nFor example, the square roots of\n−4\nare\n2i\nand\n−2i\n, and the square roots of\ni\nare\nIf we express a complex number in [polar form](https://bliptext.com/articles/polar-form), then the square root can be obtained by taking the square root of the radius and halving the angle:\nA principal root of a complex number may be chosen in various ways, for example\nwhich introduces a [branch cut](https://bliptext.com/articles/branch-cut) in the [complex plane](https://bliptext.com/articles/complex-plane) along the [positive real axis](https://bliptext.com/articles/positive-real-axis) with the condition\n0 ≤ θ < 2π\n, or along the negative real axis with\n−π < θ ≤ π\n.\nUsing the first(last) branch cut the principal square root maps to the half plane with non-negative imaginary(real) part.\nThe last branch cut is presupposed in mathematical software like [Matlab](https://bliptext.com/articles/matlab) or [Scilab](https://bliptext.com/articles/scilab).\n# [The square roots of i | upload.wikimedia.org/wikipedia/commons/3/32/Imaginary2Root.svg]\n\n## Roots of unity\nThe number 1 has n different nth roots in the complex plane, namely\nwhere\nThese roots are evenly spaced around the [unit circle](https://bliptext.com/articles/unit-circle) in the complex plane, at angles which are multiples of 2\\pi/n.\nFor example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, i, −1, and -i.\n# [The three 3rd roots of 1 | upload.wikimedia.org/wikipedia/commons/3/39/3rd///roots///of///unity.svg]\n\n## nth roots\nEvery complex number has n different nth roots in the complex plane.\nThese are\nwhere η is a single nth root, and 1, ω, ω2, ... ωn−1 are the nth roots of unity.\nFor example, the four different fourth roots of 2 are\nIn [polar form](https://bliptext.com/articles/polar-form), a single nth root may be found by the formula\nHere r is the magnitude (the modulus, also called the [absolute value](https://bliptext.com/articles/absolute-value)) of the number whose root is to be taken; if the number can be written as a+bi then.\nAlso, \\theta 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 and\nThus finding nth roots in the complex plane can be segmented into two steps.\nFirst, the magnitude of all the nth roots is the nth root of the magnitude of the original number.\nSecond, the angle between the positive horizontal axis and a ray from the origin to one of the nth roots is \\theta / n, where \\theta is the angle defined in the same way for the number whose root is being taken.\nFurthermore, all n of the nth roots are at equally spaced angles from each other.\nIf n is even, a complex number's nth roots, of which there are an even number, come in [additive inverse](https://bliptext.com/articles/additive-inverse) pairs, so that if a number r1 is one of the nth roots then r2 = –r1 is another.\nThis 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.\nAs with square roots, the formula above does not define a [continuous function](https://bliptext.com/articles/continuous-function) over the entire complex plane, but instead has a [branch cut](https://bliptext.com/articles/branch-cut) at points where θ / n is discontinuous.\n\n# Solving polynomials\nIt was once [conjectured](https://bliptext.com/articles/conjecture) that all [[polynomial](https://bliptext.com/articles/polynomial) equations](https://bliptext.com/articles/[polynomial](https://bliptext.com/articles/polynomial)-equation) could be [solved algebraically](https://bliptext.com/articles/algebraic-solution) (that is, that all roots of a [polynomial](https://bliptext.com/articles/polynomial) could be expressed in terms of a finite number of radicals and [elementary operations](https://bliptext.com/articles/elementary-arithmetic)).\nHowever, while this is true for third degree polynomials ([cubics](https://bliptext.com/articles/cubic-function)) and fourth degree polynomials ([quartics](https://bliptext.com/articles/quartic-function)), the [Abel–Ruffini theorem](https://bliptext.com/articles/abel-ruffini-theorem) (1824) shows that this is not true in general when the degree is 5 or greater.\nFor example, the solutions of the equation\nx^5 = x + 1\ncannot be expressed in terms of radicals.\n(cf. [quintic equation](https://bliptext.com/articles/quintic-equation))\n\n# Proof of irrationality for non-perfect nth power x\nAssume that \\sqrt[n]{x} is rational.\nThat is, it can be reduced to a fraction \\frac{a}{b}, where a and b are integers without a common factor.\nThis means that.\nSince x is an integer, a^nand b^nmust share a common factor if b \\neq 1.\nThis means that if b \\neq 1, is not in simplest form.\nThus b should equal 1.\nSince 1^n = 1 and,.\nThis means that x = a^n and thus,.\nThis implies that \\sqrt[n]{x} is an integer.\nSince x is not a perfect nth power, this is impossible.\nThus \\sqrt[n]{x} is irrational.","revisionCount":0},"hasMore":false,"nextCursor":null}