Nth root - Citizendia

In mathematics, an nth root of a number a is a number b such that bⁿ=a. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. 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. In Mathematics, the real numbers may be described informally in several different ways The musical instrument is spelled Cymbal. A symbol is something --- such as an object, Picture, written word a sound a piece Note that if n is even, negative numbers will not have a principal nth root. In Mathematics, the parity of an object states whether it is even or odd A negative number is a Number that is less than zero, such as −2 When n = 2, the nth root is called the square root, and when n = 3, the nth root is called the cube root. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Mathematics, a cube root of a number denoted \sqrt{x} or x1/3 is a number a such that a 3 = x

1 Symbol
2 Fundamental operations
3 Working with surds
4 Infinite series
5 Finding all roots
- 5.1 Positive real numbers
6 Solving polynomials
7 See also
- 7.1 External links
8 References

Symbol

The origin of the root symbol $\sqrt{\,\,}$ is largely speculative, some proved 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 (Jathr, in Arabic means root). Origin refers to the beginning starting-point cause or ultimate source from which a thing is derived The araB gene Promoter is a bacterial promoter activated by e L-arabinose binding Abū al-Hasan ibn ʿAlī al-Qalaṣādī (1412 in Baza, Spain &ndash 1486 in Béja, Tunisia) was an Arab Muslim mathematician Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language

But many, including Leonhard Euler,^[1] believe it originates from the letter r, the first letter of the Latin word radix which refers to the same mathematical operation. A letter is an element in an Alphabetic system of writing such as the Greek alphabet and its descendants Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. In mathematical numeral systems, the base or radix is usually the number of unique digits, including zero that a positional Numeral In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values 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. Printing is a process for reproducing text and image typically with ink on Paper using a printing press A vinculum is a horizontal line placed over a Mathematical expression, used to indicate that it is to be considered a group In Astronomy, Geography, Geometry and related sciences and contexts a plane is said to be horizontal at a given point if it is locally The German people (Deutsche are an Ethnic group, in the sense of sharing a common German culture, descent and speaking the German language as

Fundamental operations

Operations with radicals are given by the following formulas:

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

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

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

where a and b are positive. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information

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. In Linear algebra, the null vector or zero vector is the vector (0 0 &hellip 0 in Euclidean space, all of whose components are zero Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The nth roots of unity are of particular importance. In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power

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. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object Addition is the mathematical process of putting things together Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract The term "concept" is traced back to 1554–60 ( l conceptum - something conceived but what is today termed "the classical theory of concepts" is the theory of Aristotle

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

If you understand how to simplify one radical expression, then addition and subtraction is simply a question of "grouping like terms". In mathematics the word expression is a term for any well-formed combination of mathematical symbols

For example,

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

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

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

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

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. ^[2]

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. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. Notationally, the radical symbol ( $\sqrt{\,\,}$ ) depicts surds, with the upper line above the expression called the vinculum. See also Table of mathematical symbols Mathematical notation is used in Mathematics, and throughout the Physical sciences, Engineering A cube root takes the form:

$\sqrt[3]{a}$ , which corresponds to $a^{\frac{1}{3}}$ , when expressed using indices. The word index is used in variety of senses in Mathematics. In perhaps the most frequent sense an index is a Superscript

All roots can remain in surd form.

Basic techniques for working with surds arise from identities. In Mathematics, the term identity has several different important meanings An identity is an equality that remains true regardless of the values of Some basic examples include:

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

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

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

$(\frac{a}\sqrt{b})(\frac\sqrt{b}\sqrt{b}) = \frac{{a}\sqrt{b}}{b}$

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

The last of these may serve to rationalize the denominator of an expression, moving surds from the denominator to the numerator. A ratio is an expression which compares quantities relative to each other Numerator may refer to A numeral used to indicate a count particularly of the equal parts in a fraction For example in 3/4 3 is 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. Case analysis is one of the most general and applicable methods of analytical thinking depending only on the division of a problem decision or situation into a sufficient number of separate In Mathematics, the difference of two squares is when a number is squared, or multiplied by itself and is then subtracted from another squared number Variants for cube and other roots exist, as do more general formulae based on finite geometric series. In Mathematics, a geometric series is a series with a constant ratio between successive terms.

Infinite series

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

$(1+x)^{s/t} = \sum_{n=0}^\infty \frac{\displaystyle\prod_{k=0}^n (s+t-kt)}{(s+t)n!t^n}x^n$

with $\ |x|<1$ . In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness

Finding all roots

All the roots of any number, real or complex, may be found with a simple algorithm. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation The number should first be written in the form ae^iφ (see Euler's formula). This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic 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. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, an elementary matrix is a simple matrix which differs from the Identity matrix in a minimal way The Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general solution in radicals to Polynomial equations of For example, the solutions of the equation

$\ x^5=x+1$

cannot be expressed in terms of radicals.

For solving any equation of the n^th degree, see Root-finding algorithm. A root-finding algorithm is a numerical method or Algorithm, for finding a value x such that f ( x) = 0 for a given function

References

^ Leonhard Euler (1755). The principal ''n''th root \sqrt{A} of a positive Real number A, is the positive real solution of the equation x^n = The shifting nth-root algorithm is an Algorithm for extracting the ''n''th root of a positive Real number which proceeds iteratively by shifting in In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Mathematics, a cube root of a number denoted \sqrt{x} or x1/3 is a number a such that a 3 = x The twelfth root of two or \sqrt{2} is an algebraic Irrational number, representing the Frequency Ratio between any two consecutive In Mathematics, the super-root is one of the two inverse functions of Tetration. Institutiones calculi differentialis (in Latin).
^ Earliest Known Uses of Some of the Words of Mathematics. Mathematics Pages by Jeff Miller. Retrieved on 2007-04-20. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1303 - The University of Rome La Sapienza is instituted by Pope Boniface VIII.

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