Editing Nth root (section)

==Definition and notation==
[[File:NegativeOne4Root.svg|thumb|The four 4th roots of −1,<br /> none of which are real]]
[[File:NegativeOne3Root.svg|thumb|The three 3rd roots of −1,<br /> one of which is a negative real]]
An  ''{{mvar|n}}th root'' of a number ''x'', where ''n'' is a positive integer, is any of the ''n'' real or complex numbers ''r'' whose ''n''th power is ''x'':

<math display="block">r^n = x.</math>

Every positive [[real number]] ''x'' has a single positive ''n''th root, called the [[principal value|principal ''n''th root]], which is written <math>\sqrt[n]{x}</math>. For ''n'' equal to 2 this is called the principal square root and the  ''n'' is omitted. The ''n''th root can also be represented using [[exponentiation]] as ''x''{{sup|1/n}}.

For even values of ''n'', positive numbers also have a negative ''n''th root, while negative numbers do not have a real ''n''th root.  For odd values of ''n'', every negative number ''x'' has a real negative ''n''th root.  For example, −2 has a real 5th root, <math>\sqrt[5]{-2} = -1.148698354\ldots</math> but −2 does not have any real 6th roots.

Every non-zero number ''x'', real or [[Complex number|complex]], has ''n'' different complex number ''n''th roots.  (In the case ''x'' is real, this count includes any real ''n''th roots.) The only complex root of 0 is 0.

The ''n''th roots of almost all numbers (all integers except the ''n''th powers, and all rationals except the quotients of two ''n''th powers) are [[irrational number|irrational]].  For example,

<math display="block">\sqrt{2} = 1.414213562\ldots</math>

All ''n''th roots of rational numbers are [[algebraic number]]s, and all ''n''th roots of integers are [[algebraic integer]]s.

The term "surd" traces back to [[Al-Khwarizmi]] ({{circa|825}}), who referred to rational and irrational numbers as ''audible'' and ''inaudible'', respectively. This later led to the Arabic word {{lang|ar|أصم}} ({{lang|ar-Latn|asamm}}, meaning "deaf" or "dumb") for ''irrational number'' being translated into Latin as {{lang|la|surdus}} (meaning "deaf" or "mute"). [[Gerard of Cremona]] ({{circa|1150}}), [[Fibonacci]] (1202), and then [[Robert Recorde]] (1551) all used the term to refer to ''unresolved irrational roots'', that is, expressions of the form <math>\sqrt[n]{r}</math>, in which <math>n</math> and <math>r</math> are integer numerals and the whole expression denotes an irrational number.<ref>{{cite web |url=http://jeff560.tripod.com/s.html |title=Earliest Known Uses of Some of the Words of Mathematics|website=Mathematics Pages |first=Jeff |last=Miller|access-date=2008-11-30}}</ref> Irrational numbers of the form <math>\pm\sqrt{a},</math> where <math>a</math> is rational, are called ''pure quadratic surds''; irrational numbers of the form  <math>a \pm\sqrt{b}</math>, where <math>a</math> and <math>b</math> are rational, are called ''[[Quadratic irrational number|mixed quadratic surds]]''.<ref>{{cite book |last=Hardy |first=G. H. |author-link=G. H. Hardy |at=§1.13 "Quadratic Surds" – §1.14, {{pgs|19–23}} |url=https://archive.org/details/coursepuremath00hardrich/page/n36/mode/2up |title=A Course of Pure Mathematics |year=1921 |edition=3rd |publisher=Cambridge}}</ref>

===Square roots===
{{Main article|Square root}}
[[Image:Square-root function.svg|thumb|right|The graph <math>y=\pm \sqrt{x}</math>.]]
A '''square root''' of a number ''x'' is a number ''r'' which, when [[square (algebra)|squared]], becomes ''x'':

<math display="block">r^2 = x.</math>

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:

<math display="block">\sqrt{25} = 5.</math>

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 number|imaginary]] square roots.  For example, the square roots of −25 are 5''i'' and −5''i'', where ''[[imaginary unit|i]]'' represents a number whose square is {{math|−1}}.

===Cube roots===
{{Main article|Cube root}}
[[Image:cube-root function.svg|thumb|right|The graph <math>y=\sqrt[3]{x}</math>.]]
A '''cube root''' of a number ''x'' is a number ''r'' whose [[cube (algebra)|cube]] is ''x'':

<math display="block">r^3 = x.</math>

Every real number ''x'' has exactly one real cube root, written <math>\sqrt[3]{x}</math>.  For example,

<math display="block">\begin{align}
\sqrt[3]{8} &= 2\\
\sqrt[3]{-8} &= -2.
\end{align}</math>

Every real number has two additional [[complex number|complex]] cube roots.