Radical expression

In mathematics, an nth root of a number x is the number r which, when multiplied by itself n times, yields x: \[r^n = \underbrace{r \times r \times \dotsb \times r}_{n\text{ factors}} = x.\] The positive 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 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.

The nth root of x is written as \(\sqrt[n]{x}\) using the radical symbol \(\sqrt{\phantom x}\). The square root is usually written as ⁠\(\sqrt x\)⁠, with the degree omitted. Taking the nth root of a number, for fixed ⁠\(n\)⁠, is the inverse of raising a number to the nth power, and can be written as a fractional exponent:

\[\sqrt[n]{x} = x^{1/n}.\]

For a positive real number x, \(\sqrt{x}\) denotes the positive square root of x and \(\sqrt[n]{x}\) denotes the positive real nth root. For example, 3 is a square root of 9, since 3 = 9, and −3 is also a square root of 9, since (−3) = 9. A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, ⁠\(+i\sqrt x\)⁠ and ⁠\(-i\sqrt x\)⁠, where i is the imaginary unit.

In general, any non-zero complex number has n distinct complex-valued nth roots, equally distributed around a complex circle of constant absolute value. (The nth root of 0 is zero with multiplicity 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. By convention the principal value of this function, 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. 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. 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 unresolved root, especially one using the radical symbol, is sometimes referred to as a surd or a radical. 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.

The Babylonians, as early as 1800 BCE, demonstrated numerical approximations of irrational quantities such as the square root of 2 on clay tablets, with an accuracy analogous to six decimal places, as in the tablet YBC 7289. Cuneiform tablets from Larsa include tables of square and cube roots of integers. The first to prove the irrationality of √2 was most likely the Pythagorean Hippasus. Plato in his Theaetetus, then describes how Theodorus of Cyrene (c. 400 BC) proved the irrationality of \(\sqrt3\), \(\sqrt5\), etc. up to \(\sqrt{17}\). In the first century AD, Heron of Alexandria devised an iterative method to compute the square root, which is actually a special case of the more general Newton's method.

The term surd traces back to Al-Khwarizmi (c. 825), who referred to rational and irrational numbers as "audible" and "inaudible", respectively. This 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"). Gerard of Cremona (c. 1150), 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. 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. An archaic term from the late 15th century for the operation of taking nth roots is radication, and an unresolved root is a radical.

In the fourteenth century, Jamshid al-Kashi used an iterative technique now called the Ruffini-Horner method to extract nth roots for an arbitrary n. This technique has been used since antiquity to determine square roots, then by China and Kushyar ibn Labban during the tenth century to determine cube roots. In 1665, Isaac Newton discovered the general binomial theorem, which can convert an nth root into an infinite series. Based on approach developed by François Viète, Newton devised an iterative method for solving a non-linear function of the form \(f(x) = 0\), which can be used to extract an nth root. This technique was further refined by Joseph Raphson and became known as the Newton-Raphson method. In 1690, Michel Rolle introduced the notation \(\sqrt[n]{a}\) for the nth root of the value a.

In 1629, Albert Girard proposed the fundamental theorem of algebra, but failed to produce a proof. This theorem states that every single-variable polynomial of degree n has n roots. Further, a polynomial with complex coefficients has at least one complex root. Equivalently, the theorem states that the field of complex numbers is algebraically closed. Among the notable mathematicians who worked on a proof during the 18th and 19th centuries were d'Alembert, Gauss, Bolzano, and Weierstrass, with Gauss usually being credited with the first correct proof. A consequence of this proof is that any nth root of a real or complex number will be on the complex plane.

Condensed: the full section is in Wikipedia.

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:

\[r^n = x.\]

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,

\[\sqrt{2} = 1.414213562\ldots\]

Condensed: the full section is in Wikipedia.

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:

\[\sqrt{25} = 5.\]

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 −1.

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,

\[\begin{align} \sqrt[3]{8} &= 2\\ \sqrt[3]{-8} &= -2. \end{align}\]

Every real number has two additional complex cube roots.

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,

\[\sqrt[n]{a^m} = (a^m)^{1/n} = a^{m/n} = (a^{1/n})^m = (\sqrt[n]a)^m.\]

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:

\[\begin{align} \sqrt[n]{ab} &= \sqrt[n]{a} \sqrt[n]{b} \\ \sqrt[n]{\frac{a}{b}} &= \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{align}\]

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

\[\sqrt{-1}\times\sqrt{-1} \neq \sqrt{-1 \times -1} = 1,\quad\]

but, rather,

Condensed: the full section is in Wikipedia.

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.

For example, to write the radical expression \(\textstyle \sqrt{32/5}\) in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:

\[\sqrt{\frac{32}{5}} = \sqrt{\frac{16 \cdot 2}{5}} = \sqrt{16} \cdot \sqrt{\frac{2}{5}} = 4 \sqrt{\frac{2}{5}}\]

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

\[4 \sqrt{\frac{2}{5}} = \frac{4 \sqrt{2}}{\sqrt{5}}\]

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

\[\frac{4 \sqrt{2}}{\sqrt{5}} = \frac{4 \sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{4 \sqrt{10}}{5} = \frac{4}{5}\sqrt{10}\]

Condensed: the full section is in Wikipedia.

The radical or root may be represented by the generalized binomial theorem:

\[(1+x)^{s/t} = \sum_{m=0}^\infty \frac{x^m}{m!} \prod_{k=0}^{m-1} \left(\frac st - k\right)\]

with \(|x|<1\). This expression can be derived from the binomial series. For the nth root, this becomes

\[(1+x)^\frac{1}{n} = \sum_{m=0}^\infty \frac{x^m}{m!} \prod_{k=0}^{m-1} \left(\frac{1}{n} - k\right)\]

For numbers \(r \ge 2\), choose a value \(p\) such that

\[\frac{r}{p^n} - 1 = x', \text{ where } |x'| < 1\]

then per above, solve for

Condensed: the full section is in Wikipedia.

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

\[x_{k+1} = x_k-\frac{x_k^n-A}{nx_k^{n-1}}\]

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

\[x_{k+1} = \frac{1}{n} \left[(n-1)\,x_k+\,\frac{A}{x_k^{n-1}}\right].\]

This allows the relation to only have one exponentiation, which is computed once for each iteration. The nth root of x can then be defined as the limit of \(x_k\) as k approaches infinity.

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:

(All correct digits shown.)

Condensed: the full section is in Wikipedia.

The technique of François Viète, published c. 1600, can be used to perform digit-by-digit calculation of principal roots of decimal (base 10) numbers. This method is based on the binomial theorem and is essentially an inverse algorithm solving \[(10 x+y)^n = \sum_{k=0}^n P(n, k) (10 x)^{n-k} y^k\] where \(P(n, k)\), the binomial coefficient, is the kth entry on the nth row of Pascal's triangle.

To compute the root of a number C, choose a series of approximations \[x_i^n, i = 0, 1, \ldots \text{ with } x_0 = 0\] that satisfy \(x_i^n \le C\), where the difference between \(x_{i+1}\) and \(x_i\) is the next digit in the approximation. The decimal fraction \(y_i\) is chosen to be the largest number with a single significant digit that satisfies \[10 x_i + y_i = 10 x_{i+1}, \text{ where } x_{i+1}^n \le C\] then per the binomial theorem \[(10 x_i+y)^n - (10 x_i)^n = \sum_{k=0}^{n-1} P(n, k) (10 x_i)^{n-k} y_i^k \le 10^n(C - x_i^n)\] The term \(10^n(C - x_i^n)\) is just a \(10^n\) multiple of the ith remainder, \(C - x_i^n\).

Using this 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.
2. 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 \(0^0 = 1\)).
   • Determine the greatest digit \(x\) such that \(y \le c\).
   • 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.
3. Subtract \(y\) from \(c\) to form a new remainder.
4. 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.

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

\[n \log_b r = \log_b x \quad \quad \text{hence} \quad \quad \log_b r = \frac{\log_b x}{n}.\]

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

\[r = b^{\frac{1}{n}\log_b x}.\]

(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 r = −|r|.

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

\[\tfrac{1}{\sqrt{2}}(1 + i) \quad\text{and}\quad -\tfrac{1}{\sqrt{2}}(1 + i).\]

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:

\[\sqrt{re^{i\theta}} = \pm\sqrt{r} \cdot e^{i\theta/2}.\]

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

\[\sqrt{re^{i\theta}} = \sqrt{r} \cdot e^{i\theta/2}\]

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.

Condensed: the full section is in Wikipedia.

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

\[1,\;\omega,\;\omega^2,\;\ldots,\;\omega^{n-1},\]

where

\[\omega = e^\frac{2\pi i}{n} = \cos\left(\frac{2\pi}{n}\right) + i\sin\left(\frac{2\pi}{n}\right).\]

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 \(-i\). As a result of this symmetry, the sum of the nth roots of unity equals zero. \[\sum_{k=0}^{n-1} e^{\frac{2\pi i}{n} k} = 0\]

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

\[\eta,\;\eta\omega,\;\eta\omega^2,\;\ldots,\;\eta\omega^{n-1},\]

where η is a single nth root, and 1, ω, ω, ... ω are the nth roots of unity. Thus, since they are all just multiplied by the same scalar η, the sum of the nth roots equals zero. For example, the four different fourth roots of 2 are

\[\sqrt[4]{2},\quad i\sqrt[4]{2},\quad -\sqrt[4]{2},\quad\text{and}\quad -i\sqrt[4]{2}.\]

In polar form, a single nth root may be found from Demoivre's theorem:

\[z^\frac{1}{n} = \sqrt[n]{re^{i\theta}} = \sqrt[n]{r} \cdot e^{i\theta/n} = r^\frac{1}{n} \cdot \left( \cos\left(\frac{\theta}{n}\right) + i \sin\left(\frac{\theta}{n}\right) \right)\]

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 \(a + i b\) then \(r=\sqrt{a^2+b^2}\). The \(\theta\) 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

\(\cos \theta = \frac{a}{r}, \sin \theta = \frac{b}{r}, \text{ and } \tan \theta = \frac{b}{a}.\)

\(\sqrt[n]{r} \cdot \left( \cos\left(\frac{\theta + 2\pi k}{n}\right) + i \sin\left(\frac{\theta + 2\pi k}{n}\right) \right) \text{ for } k = 0, 1, 2, \ldots, n - 1.\)

Condensed: the full section is in Wikipedia.