nth root

In mathematics, an nth root of a number a is a number b such that bⁿ=a. When referring to the nth root of a real number a it is assumed that what is desired is the principal nth root of the number, which is denoted ${\sqrt[{n}]{a}}$ using the radical symbol ${\sqrt {\,\,}}$. The principal nth root of a real number a is the unique real number b which is an nth root of a and is of the same sign as a. Note that if n is even, negative numbers will not have a principal nth root. When n = 2, the nth root is called the square root, and when n = 3, the nth root is called the cube root.

Symbol

The origin of the root symbol √ is largely speculative. Some sources tell that the symbol was first used by Arabs, the first known use was by Abū al-Hasan ibn Alī al-Qalasādī (1421-1486), and that it is taken from the Arabic letter ج, the first letter in the word (Jathir, [with the "th" pronounced like the "th" in the english word "the"] in Arabic means root).

But many, including Leonhard Euler, believe it originates from the letter r, the first letter of the Latin word radix which refers to the same mathematical operation. The symbol was first seen in print without the vinculum (the horizontal bar over the numbers inside the radical symbol) in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician.

Fundamental operations

Operations with radicals are given by the following formulas:

${\sqrt[{n}]{ab}}={\sqrt[{n}]{a}}{\sqrt[{n}]{b}}\qquad a\geq 0,b\geq 0$

${\sqrt[{n}]{\frac {a}{b}}}={\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}}\qquad a\geq 0,b>0$

${\sqrt[{n}]{a^{m}}}=\left({\sqrt[{n}]{a}}\right)^{m}=\left(a^{\frac {1}{n}}\right)^{m}=a^{\frac {m}{n}},$

where a and b are positive.

For every non-zero complex number a, there are n different complex numbers b such that bⁿ = a, so the symbol ${\sqrt[{n}]{a}}$ cannot be used unambiguously. The nth roots of unity are of particular importance.

Once a number has been changed from radical form to exponentiated form, the rules of exponents still apply (even to fractional exponents), namely

$a^{m}a^{n}=a^{m+n}\,$

$\left({\frac {a}{b}}\right)^{m}={\frac {a^{m}}{b^{m}}}$

$(a^{m})^{n}=a^{mn}\,$

For example:

${\sqrt[{3}]{a^{5}}}{\sqrt[{5}]{a^{4}}}=a^{\frac {5}{3}}a^{\frac {4}{5}}=a^{\frac {25+12}{15}}=a^{\frac {37}{15}}$

${\frac {\sqrt {a}}{\sqrt[{4}]{a}}}=a^{\frac {1}{2}}a^{\frac {-1}{4}}=a^{\frac {4-2}{8}}=a^{\frac {2}{8}}=a^{\frac {1}{4}}$

If you are going to do addition or subtraction, then you should notice that the following concept is important.

${\sqrt[{3}]{a^{5}}}={\sqrt[{3}]{aaaaa}}={\sqrt[{3}]{a^{3}a^{2}}}=a{\sqrt[{3}]{a^{2}}}$

If you understand how to simplify one radical expression, then addition and subtraction is simply a question of "grouping like terms".

For example,

${\sqrt[{3}]{a^{5}}}+{\sqrt[{3}]{a^{8}}}$

$={\sqrt[{3}]{a^{3}a^{2}}}+{\sqrt[{3}]{a^{6}a^{2}}}$

$=a{\sqrt[{3}]{a^{2}}}+a^{2}{\sqrt[{3}]{a^{2}}}$

$=({a+a^{2}}){\sqrt[{3}]{a^{2}}}$

Working with surds

Surd

al-Khowarizmi (c. 825) referred to rational and irrational numbers as 'audible' and 'inaudible', respectively. This later lead to the Arabic "asamm" (deaf, dumb) for irrational number being translated as surdus ("deaf" or "mute") into Latin. Gherardo of Cremona (c. 1150), Fibonacci (1202) and then Robert Recorde (1551) used the term to refer to unresolved irrational roots.

Often it is simpler to leave the nth roots of numbers "unresolved" (ie. with radicals visible). These unresolved expressions, called "surds", may then be manipulated into simpler forms or arranged to divide each other out. Notationally, the radical symbol (${\sqrt {\,\,}}$) depicts surds, with the upper line above the expression called the vinculum. A cube root takes the form:

${\sqrt[{3}]{a}}$, which corresponds to $a^{\frac {1}{3}}$, when expressed using indices.

All roots can remain in surd form.

Basic techniques for working with surds arise from identities. Some basic examples include:

• ${\sqrt {a^{2}b}}=a{\sqrt {b}}$
  • The above can be combined with index reduction: ${\sqrt[{6}]{a^{6}b^{4}}}={\sqrt[{3\cdot 2}]{a^{2}a^{2}a^{2}b^{2}b^{2}}}={\sqrt[{3}]{a^{3}b^{2}}}=a{\sqrt[{3}]{b^{2}}}$
• ${\sqrt[{n}]{a^{m}b}}=a^{\frac {m}{n}}{\sqrt[{n}]{b}}$

• ${\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}$

• ${\frac {\sqrt {a}}{\sqrt {b}}}={\sqrt {\frac {a}{b}}}$

• $\left({\frac {a}{\sqrt {b}}}\right)\left({\frac {\sqrt {b}}{\sqrt {b}}}\right)={\frac {{a}{\sqrt {b}}}{b}}$

• $({\sqrt {a}}+{\sqrt {b}})^{-1}={\frac {1}{({\sqrt {a}}+{\sqrt {b}})}}={\frac {{\sqrt {a}}-{\sqrt {b}}}{({\sqrt {a}}+{\sqrt {b}})({\sqrt {a}}-{\sqrt {b}})}}={\frac {{\sqrt {a}}-{\sqrt {b}}}{a-b}}.$

The last of these may serve to rationalize the denominator of an expression, moving surds from the denominator to the numerator. It follows from the identity

$({\sqrt {a}}+{\sqrt {b}})({\sqrt {a}}-{\sqrt {b}})=a-b$,

which exemplifies a case of the difference of two squares. Variants for cube and other roots exist, as do more general formulae based on finite geometric series.

Infinite series

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

$(1+x)^{s/t}=\sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(s-kt)}{n!t^{n}}}x^{n}$

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

Computing principal roots

The nth root of an integer is in general not an integer or rational number. For instance, the fifth root of 34 is

${\sqrt[{5}]{34}}=2.024397458\ldots$

The nth root of a number A can be computed by the nth root algorithm. Start with an initial guess x₀ and then iterate using the recurrence relation

$x_{k+1}={\frac {1}{n}}\left({(n-1)x_{k}+{\frac {A}{x_{k}^{n-1}}}}\right)$

until the desired precision is reached.

Another method is to use the infinite series mentioned in the previous section. Depending on the application, it may be enough to use only the two terms in this series:

${\sqrt[{n}]{x+y}}\approx {\sqrt[{n}]{x}}+{\frac {y}{n\left({\sqrt[{n}]{x}}\right)^{n-1}}}.$

For example, to find the fifth root of 34, note that 2⁵ = 32 and thus take x = 32 and y = 2 in the above formula. This yields

${\sqrt[{5}]{34}}={\sqrt[{5}]{32+2}}\approx 2+{\frac {2}{5\cdot 16}}=2.025.$

The error in the approximation is only about 0.03 %.

Finding all the roots of a given number

All the roots of any number, real or complex, may be found with a simple algorithm. The number should first be written in the form ae^iφ (the so-called polar form). Then all the nth roots are given by:

$e^{({\frac {\phi +2\pi k}{n}})i}\times {\sqrt[{n}]{a}}$

for $k=0,1,2,\ldots ,n-1$, where ${\sqrt[{n}]{a}}$ represents the principal nth root of a.

Positive real numbers

All the complex solutions of xⁿ = a, or the nth roots of a, where a is a positive real number, are given by the simplified equation:

$e^{2\pi i{\frac {k}{n}}}\times {\sqrt[{n}]{a}}$

for $k=0,1,2,\ldots ,n-1$, where ${\sqrt[{n}]{a}}$ represents the principal nth root of a.

Solving polynomials

It was once conjectured that all roots of polynomials could be expressed in terms of radicals and elementary operations; however, the Abel-Ruffini theorem asserts that this is not true in general. For example, the solutions of the equation

x⁵ = x + 1

cannot be expressed in terms of radicals.

For solving any equation of the n^th degree, see Root-finding algorithm.

See also

• Nth root algorithm
• Shifting nth-root algorithm
• Irrational number
• Algebraic number
• Square root
• Cube root
• Twelfth root of two
• Super-root

References

External links

• principal nth root calculator reduces any number to principal nth root, shows simplest radical form
• General High-order nth-root algorithms