Nth root explained

In mathematics, an th root of a number is a number (the root) which, when raised to the power of the positive integer, yields : $$r^n\; =\; \backslash underbrace\_\; =\; x.$$

The integer is called the index or degree, and the number of which the root is taken is the radicand. 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, is a square root of, since, and is also a square root of, since .

The th root of is written as

\sqrt[n]{x} using the radical symbol or radix \sqrt{\phantom x} . The square root is usually written without the as just . Taking the nth root of a number is the inverse operation of exponentiation,^[1] and can be written as a fractional exponent:

$$\backslash sqrt[n]\; =\; x^.$$

For a positive real number,

\sqrt{x} denotes the positive square root of and \sqrt[n]{x} denotes the positive real th root. A negative real number has no real-valued square roots, but when is treated as a complex number it has two imaginary square roots, and, where is the imaginary unit.

In general, any non-zero complex number has distinct complex-valued th roots, equally distributed around a complex circle of constant absolute value. (The th root of is zero with multiplicity, and this circle degenerates to a point.) Extracting the th roots of a complex number can thus be taken to be a multivalued function. By convention the principal value of this function, called the principal root and denoted, is taken to be the th root with the greatest real part and in the special case when is a negative real number, the one with a positive imaginary part. The principal root of a positive real number is thus also a positive real number. As a function, the principal root is continuous in the whole complex plane, except along the negative real axis.

An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd^[2] or a radical.^[3] 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 are used for determining the radius of convergence of a power series with the root test. The th 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.

History

See main article: article. An archaic term for the operation of taking nth roots is radication.^{[4] [5]}

Definition and notation

An th 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:

$$r^n\; =\; x.$$

Every positive real number x has a single positive nth root, called the principal nth root, which is written

\sqrt[n]{x} . For n equal to 2 this is called the principal square root and the n is omitted. The nth root can also be represented using exponentiation as x.

For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nth root. For odd values of n, every negative number x has a real negative nth root. For example, −2 has a real 5th root,

\sqrt[ 5 ]{ -2 } = -1 . 148698354 \ldots but −2 does not have any real 6th roots.

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,

$$\backslash sqrt\; =\; 1.414213562\backslash ldots$$

All nth roots of rational numbers are algebraic numbers, and all nth roots of integers are algebraic integers.

The term "surd" traces back to Al-Khwarizmi, who referred to rational and irrational numbers as audible and inaudible, respectively. This later led to the Arabic word Arabic: أصم (Arabic: asamm, meaning "deaf" or "dumb") for irrational number being translated into Latin as Latin: surdus (meaning "deaf" or "mute"). Gerard of Cremona, Fibonacci (1202), and then 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.^[6] Irrational numbers of the form \pm\sqrt{a}, where a is rational, are called pure quadratic surds; irrational numbers of the form a \pm\sqrt{b} , where a and b are rational, are called mixed quadratic surds.^[7]

Square roots

See main article: article and Square root. A square root of a number x is a number r which, when squared, becomes x:

$$r^2\; =\; x.$$

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:

$$\backslash sqrt\; =\; 5.$$

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 .

Cube roots

See main article: article and Cube root. A cube root of a number x is a number r whose cube is x:

$$r^3\; =\; x.$$

Every real number x has exactly one real cube root, written

\sqrt[ 3 ]{x} . For example,

Every real number has two additional complex cube roots.

Identities and properties

Expressing the degree of an nth root in its exponent form, as in

x ^{1/n} , makes it easier to manipulate powers and roots. If a is a non-negative real number,

$$\backslash sqrt[n]\; =\; (a^m)^\; =\; a^\; =\; (a^)^m\; =\; (\backslash sqrt[n]a)^m.$$

Every 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:

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

$$\backslash sqrt\backslash times\backslash sqrt\; \backslash neq\; \backslash sqrt\; =\; 1,\backslash quad$$

but, rather,

$$\backslash quad\backslash sqrt\backslash times\backslash sqrt\; =\; i\; \backslash times\; i\; =\; i^2\; =\; -1.$$

Since the rule

\sqrt[n]{a} x \sqrt[n]{b} = \sqrt[n]{ab} 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 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.^[8]

For example, to write the radical expression

style \sqrt{ 32 / 5 } in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:

$$\backslash sqrt\; =\; \backslash sqrt=\; \backslash sqrt\; \backslash cdot\; \backslash sqrt\; =\; 4\; \backslash sqrt$$

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

$$4\; \backslash sqrt\; =\; \backslash frac$$

Finally, we remove the radical from the denominator as follows:

$$\backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\backslash sqrt$$

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.^{[9] [10]} For instance using the factorization of the sum of two cubes:

$$\backslash frac\; =\; \backslash frac\; =\; \backslash frac\; .$$

Simplifying radical expressions involving nested radicals can be quite difficult. In particular, denesting is not always possible, and when possible, it may involve advanced Galois theory. Moreover, when complete denesting is impossible, there is no general canonical form such that the equality of two numbers can be tested by simply looking at their canonical expressions.

For example, it is not obvious that

$$\backslash sqrt\; =\; 1\; +\; \backslash sqrt.$$

The above can be derived through:

$$\backslash sqrt\; =\; \backslash sqrt\; =\; \backslash sqrt\; =\; \backslash sqrt\; =\; 1\; +\; \backslash sqrt$$

Let

r = p/q , with and coprime and positive integers. Then \sqrt[n]r = \sqrt[n]{p}/\sqrt[n]{q} is rational if and only if both \sqrt[n]{p} and \sqrt[n]{q} are integers, which means that both and are nth powers of some integer.

Infinite series

The radical or root may be represented by the infinite series:

$$(1+x)^\backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash fracx^n$$

with

|x|< 1 . This expression can be derived from the binomial series.

Computing principal roots

Using Newton's method

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

$$x\_\; =\; x\_k-\backslash frac$$

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

$$x\_\; =\; \backslash frac\backslash ,x\_k+\backslash frac\backslash ,\backslash frac\; 1.$$

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 and (initial guess). The first 5 iterations are, approximately:

(All correct digits shown.)

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

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

$$\backslash sqrt[n]\; =\; \backslash sqrt[n]\; =\; x+\backslash cfrac\; .$$

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,

x( 20 p + x) \le c , or x ^{2} + 20 xp \le c , follows a pattern involving Pascal's triangle. For 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 \sum ^{n -1} _{i =0} 10 ^{i} P(n,i)p ^{i} x ^{n-i} . For 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.

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:

1. Starting on the left, bring down the most significant (leftmost) group of digits not yet used (if all the digits have been used, write "0" the number of times required to make a group) and write them to the right of the remainder from the previous step (on the first step, there will be no remainder). In other words, multiply the remainder by

10 ^{n} and add the digits from the next group. This will be the current value c.

1. Find p and x, as follows:
   • Let

p be the part of the root found so far, ignoring any decimal point. (For the first step, p = 0 and 0 ^{0} = 1 ).

1. • Determine the greatest digit

x such that y \le c .

1. • Place the digit

x as the next digit of the root, i.e., above the group of digits you just brought down. Thus the next p will be the old p times 10 plus x.

1. Subtract

y from c to form a new remainder.

1. If the remainder is zero and there are no more digits to bring down, then the algorithm has terminated. Otherwise go back to step 1 for another iteration.

Examples

Find the square root of 152.2756.

1 2. 3 4 / \/ 01 52.27 56 (Results) (Explanations)   01 x = 1 10·1·0·1 + 10·2·0·1 ≤ 1 < 10·1·0·2 + 10·2·0·2 01 y = 1 y = 10·1·0·1 + 10·2·0·1 = 1 + 0 = 1 00 52 x = 2 10·1·1·2 + 10·2·1·2 ≤ 52 < 10·1·1·3 + 10·2·1·3 00 44 y = 44 y = 10·1·1·2 + 10·2·1·2 = 4 + 40 = 44 08 27 x = 3 10·1·12·3 + 10·2·12·3 ≤ 827 < 10·1·12·4 + 10·2·12·4 07 29 y = 729 y = 10·1·12·3 + 10·2·12·3 = 9 + 720 = 729 98 56 x = 4 10·1·123·4 + 10·2·123·4 ≤ 9856 < 10·1·123·5 + 10·2·123·5 98 56 y = 9856 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 truncated to the nearest thousandth.

1 6. 1 2 4 3 / \/ 004 192.000 000 000 (Results) (Explanations)   004 x = 1 10·1·0·1 + 10·3·0·1 + 10·3·0·1 ≤ 4 < 10·1·0·2 + 10·3·0·2 + 10·3·0·2 001 y = 1 y = 10·1·0·1 + 10·3·0·1 + 10·3·0·1 = 1 + 0 + 0 = 1 003 192 x = 6 10·1·1·6 + 10·3·1·6 + 10·3·1·6 ≤ 3192 < 10·1·1·7 + 10·3·1·7 + 10·3·1·7 003 096 y = 3096 y = 10·1·1·6 + 10·3·1·6 + 10·3·1·6 = 216 + 1,080 + 1,800 = 3,096 096 000 x = 1 10·1·16·1 + 10·3·16·1 + 10·3·16·1 ≤ 96000 < 10·1·16·2 + 10·3·16·2 + 10·3·16·2 077 281 y = 77281 y = 10·1·16·1 + 10·3·16·1 + 10·3·16·1 = 1 + 480 + 76,800 = 77,281 018 719 000 x = 2 10·1·161·2 + 10·3·161·2 + 10·3·161·2 ≤ 18719000 < 10·1·161·3 + 10·3·161·3 + 10·3·161·3 015 571 928 y = 15571928 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 x = 4 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

The desired precision is achieved. The cube root of 4192 is 16.124...

Logarithmic calculation

The principal nth root of a positive number can be computed using logarithms. Starting 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 will do) to obtain

$$n\; \backslash log\_b\; r\; =\; \backslash log\_b\; x\; \backslash quad\; \backslash quad\; \backslash text\; \backslash quad\; \backslash quad\; \backslash log\_b\; r\; =\; \backslash frac.$$

The root r is recovered from this by taking the antilog:

$$r\; =\; b^.$$

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

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

|r| ^{n} = |x|, 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 nth 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 nth 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

$$\backslash tfrac(1\; +\; i)\; \backslash quad\backslash text\backslash quad\; -\backslash tfrac(1\; +\; i).$$

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:

$$\backslash sqrt\; =\; \backslash pm\backslash sqrt\; \backslash cdot\; e^.$$

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

$$\backslash sqrt\; =\; \backslash sqrt\; \backslash cdot\; e^$$

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

\scriptstyle \sqrt z maps \scriptstyle z 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

See main article: article and Root of unity.

The number 1 has n different nth roots in the complex plane, namely

$$1,\backslash ;\backslash omega,\backslash ;\backslash omega^2,\backslash ;\backslash ldots,\backslash ;\backslash omega^,$$

where

$$\backslash omega\; =\; e^\backslash frac\; =\; \backslash cos\backslash left(\backslash frac\backslash right)\; +\; i\backslash sin\backslash left(\backslash frac\backslash right).$$

These roots are evenly spaced around the unit circle in the complex plane, at angles which are multiples of

2 \pi/n . For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, i , −1, and - i .

nth roots

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

$$\backslash eta,\backslash ;\backslash eta\backslash omega,\backslash ;\backslash eta\backslash omega^2,\backslash ;\backslash ldots,\backslash ;\backslash eta\backslash omega^,$$

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

$$\backslash sqrt[4],\backslash quad\; i\backslash sqrt[4],\backslash quad\; -\backslash sqrt[4],\backslash quad\backslash text\backslash quad\; -i\backslash sqrt[4].$$

In polar form, a single nth root may be found by the formula

$$\backslash sqrt[n]\; =\; \backslash sqrt[n]\; \backslash cdot\; e^.$$

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

r = \sqrt{a ^{2+b} ^{2}} . Also, \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 \cos \theta = a/r, \sin \theta = b/r, and \tan \theta = b/a.

Thus finding nth roots in the complex plane can be segmented into two steps. First, the magnitude of all the nth roots is the nth 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 nth roots is

\theta / n , where \theta is the angle defined in the same way for the number whose root is being taken. Furthermore, all n of the nth roots are at equally spaced angles from each other.

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) = (–1) × r₁ = r₁.

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

See also: Root-finding algorithms.

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

$$x^5\; =\; x\; +\; 1$$

cannot be expressed in terms of radicals. (cf. quintic equation)

Proof of irrationality for non-perfect nth power x

Assume that

\sqrt[n]{x} is rational. That is, it can be reduced to a fraction (a)/(b) , where and are integers without a common factor.

This means that

x = (a ^{n})/(b ^{n}) .

Since x is an integer,

a ^{n} and b ^{n} must share a common factor if b ≠ 1 . This means that if b ≠ 1 , (a ^{n})/(b ^{n}) is not in simplest form. Thus b should equal 1.

Since

1 ^{n} = 1 and (n)/(1) = n , (a ^{n})/(b ^{n}) = a ^{n} .

This means that

x = a ^{n} and thus, \sqrt[n]{x} = a . This implies that \sqrt[n]{x} is an integer. Since is not a perfect th power, this is impossible. Thus \sqrt[n]{x} is irrational.

See also

• Geometric mean
• Twelfth root of two

Notes and References

1. Web site: Lesson Explainer: nth Roots: Integers. 22 July 2023 .
2. Book: Bansal, R.K. . New Approach to CBSE Mathematics IX . 25 . 2006 . 978-81-318-0013-3 . Laxmi Publications .
3. Book: Silver, Howard A.. Algebra and trigonometry. 1986. Prentice-Hall. Englewood Cliffs, New Jersey. 978-0-13-021270-2. registration.
4. Web site: Definition of RADICATION. www.merriam-webster.com.
5. Web site: radication – Definition of radication in English by Oxford Dictionaries. https://web.archive.org/web/20180403112348/https://en.oxforddictionaries.com/definition/radication. dead. April 3, 2018. Oxford Dictionaries .
6. Web site: Earliest Known Uses of Some of the Words of Mathematics. Mathematics Pages . Jeff . Miller. 2008-11-30.
7. Book: Hardy, G. H. . G. H. Hardy . §1.13 "Quadratic Surds" – §1.14, . A Course of Pure Mathematics . 1921 . 3rd . Cambridge.
8. Book: McKeague, Charles P.. Elementary algebra. 470. 2011. Cengage Learning . 978-0-8400-6421-9.
9. B. F. . Caviness. R. J. . Fateman. http://www.eecs.berkeley.edu/~fateman/papers/radcan.pdf. Simplification of Radical Expressions. Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation. 329.
10. Richard. Zippel. Simplification of Expressions Involving Radicals. Journal of Symbolic Computation. 1. 189–210. 1985. 189–210 . 10.1016/S0747-7171(85)80014-6.