- Nth root
In

mathematics , an**"n"th root**of anumber "a" is a number "b" such that "b^{n}"="a". When referring to "the" "n"th root of areal number "a" it is assumed that what is desired is the**principal "n"th root**of the number, which is denoted $sqrt\; [n]\; \{a\}$ using the**radical**symbol $sqrt\{,,\}$. The principal "n"th root of a real number "a" is the unique real number "b" which is an "n"th root of "a" and is of the same sign as "a". Note that if "n" is even,negative number s will not have a principal "n"th root. When "n" = 2, the "n"th root is called thesquare root , and when "n" = 3, the "n"th root is called thecube root .**ymbol**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 theArabic letter" ج", the first letter in the word ("Jathr", in Arabic means root).But many, including

Leonhard Euler , [*cite book|title="Institutiones calculi differentialis"|author=Leonhard Euler|year=1755|language=Latin*] believe it originates from the letter "r", the first letter of theLatin 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 year1525 in "Die Coss" byChristoff Rudolff , a German mathematician.**Fundamental operations**Operations with radicals are given by the following

formula s::$sqrt\; [n]\; \{ab\}\; =\; sqrt\; [n]\; \{a\}\; sqrt\; [n]\; \{b\}\; qquad\; a\; ge\; 0,\; b\; ge\; 0$

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

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

where "a" and "b" are

positive .For every non-zero

complex number "a", there are "n" different complex numbers "b" such that "b"^{"n"}= "a", so the symbol $sqrt\; [n]\; \{a\}$ cannot be used unambiguously. The "n"th 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

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

:$left(\{frac\{a\}\{b\; ight)^m\; =\; frac\{a^m\}\{b^m\}$

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

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 orsubtraction , then you should notice that the following concept is important.:$sqrt\; [3]\; \{a^5\}\; =\; sqrt\; [3]\; \{aaaaa\}\; =\; sqrt\; [3]\; \{a^3a^2\}\; =\; asqrt\; [3]\; \{a^2\}$If you understand how to simplify one radical expression, then addition and subtraction is simply a question of "grouping like terms".

For example,:$sqrt\; [3]\; \{a^5\}+sqrt\; [3]\; \{a^8\}$:$=sqrt\; [3]\; \{a^3a^2\}+sqrt\; [3]\; \{a^6\; a^2\}$:$=asqrt\; [3]\; \{a^2\}+a^2sqrt\; [3]\; \{a^2\}$:$=(\{a+a^2\})sqrt\; [3]\; \{a^2\}$

**Working with surds**Often it is simpler to leave the "n"th 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: $sqrt\; [6]\; \{a^6b^4\}\; =\; sqrt\; [3cdot\; 2]\; \{a^2a^2a^2b^2b^2\}\; =\; sqrt\; [3]\; \{a^3b^2\}\; =\; asqrt\; [3]\; \{b^2\}$

*$sqrt\; [n]\; \{a^m\; b\}\; =\; a^\{frac\{m\}\{nsqrt\; [n]\; \{b\}$*$sqrt\{a\}\; sqrt\{b\}\; =\; sqrt\{ab\}$

*$fracsqrt\{a\}sqrt\{b\}\; =\; sqrtfrac\{a\}\{b\}$

*$left(frac\{a\}sqrt\{b\}\; ight)left(fracsqrt\{b\}sqrt\{b\}\; ight)\; =\; frac$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 thenumerator . 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 finitegeometric 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 "n"th root of an integer is in general not an integer or rational number. For instance, the fifth root of 34 is:$sqrt\; [5]\; \{34\}\; =\; 2.024397458ldots$

The "n"th root of a number "A" can be computed by the "n"th root algorithm. Start with an initial guess "x"

_{0}and then iterate using the recurrence relation:$x\_\{k+1\}\; =\; frac\{1\}\{n\}\; left(\{(n-1)x\_k\; +frac\{A\}\{x\_k^\{n-1\}\; ight)$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::$sqrt\; [n]\; \{x+y\}\; approx\; sqrt\; [n]\; \{x\}\; +\; frac\{y\}\{nleft(sqrt\; [n]\; \{x\}\; ight)^\{n-1.$For example, to find the fifth root of 34, note that 2

^{5}= 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 "n"th roots are given by::$e^\{(frac\{phi+2pi\; k\}\{n\})i\}\; imes\; sqrt\; [n]\; \{a\}$for $k=0,1,2,ldots,n-1$, where $sqrt\; [n]\; \{a\}$ represents the principal "n"th root of "a".**Positive real numbers**All the complex solutions of "x

^{n}" = "a", or the "n"th roots of "a", where "a" is a positive real number, are given by the simplified equation::$e^\{2pi\; i\; frac\{k\}\{n\; imes\; sqrt\; [n]\; \{a\}$for $k=0,1,2,ldots,n-1$, where $sqrt\; [n]\; \{a\}$ represents the principal "n"th root of "a".**olving polynomials**It was once

conjecture d that all roots ofpolynomial s could be expressed in terms of radicals andelementary operations ; however, theAbel-Ruffini theorem asserts that this is not true in general. For example, the solutions of the equation: x^{5}= x + 1cannot be expressed in terms of radicals.For solving any equation of the n

^{th}degree, seeRoot-finding algorithm .**ee 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*** [

*http://www.mathwarehouse.com/arithmetic/principal-nth-root-calculator.php principal nth root calculator*] reduces any number to principal nth root, shows simplest radical form

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