NationMaster - Encyclopedia: Radical (mathematics)

In mathematics, an nth root of a number a is a number b, such that bⁿ=a. When referring to the n-th root of a real number a is it assumed you are talking about the principal n-th root of the number, which is denotated $sqrt[n]{a}$ . The principal square root of a real number a is the unique real number b that is an n-th root of a and satisfies the equation sign(a) = sign(b). Note that if n is even, negative numbers will not have a principal nth root. See square root for the case where n = 2. Euclid, detail from The School of Athens by Raphael. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is For example, since This example suggests how square roots can arise when solving quadratic equations such as or...

1 Fundamental operations
2 Working with surds
3 Infinite Series
4 Finding all of the roots
5 Solving polynomials
6 See also

Fundamental operations

Operations with radicals are given by the following formulas:

$sqrt[n]{ab} = sqrt[n]{a} sqrt[n]{b},$

$sqrt[n]{frac{a}{b}} = frac{sqrt[n]{a}}{sqrt[n]{b}},$

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

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. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, the nth roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n. ...

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

a m a n = a m + n

$({frac{a}{b}})^m = frac{a^m}{b^m}$

(a m) n = a m n

For example:

$sqrt[3]{a^5}sqrt[5]{a^4} = a^{5/3} a^{4/5} = a^{5/3 + 4/5} = a^{37/15}$

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

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

If you understand how to simplify one radical expression, then addition and subtraction is simply a question about 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 in calculations, it pays to leave the square or other roots of numbers unresolved. One can then manipulate these unresolved expressions into simpler forms or arrange them to cancel each other. Notationally, the √ symbol depicts surds, with an upper line above the expression (called the vinculum) enclosed in the root. A cube root takes the form: In phonetics, surd is an older (and now rarely-used) alternate name for a voiceless consonant. ... A calculation is a deliberate process for transforming one or more inputs into one or more results. ... A vinculum is a horizontal line placed over a mathematical expression, used to indicate that it is to be considered a group. ...

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

Square roots, cube roots and so on, can all remain in surd form. In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is For example, since This example suggests how square roots can arise when solving quadratic equations such as or... Plot of y = âˆ›x In mathematics, the cube root (âˆ›) of a number is a number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. ...

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

$sqrt{a^2 b} = a sqrt{b}$

$sqrt[n]{a^m b} = a^{frac{m}{n}}sqrt[n]{b}$

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

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

The last of these can serve to rationalise the denominator of an expression, moving surds from the denominator to the numerator. It follows from the identity In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...

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

which exemplifies a case of the difference of two squares. A cube root variant exists, as do more general formulae based on summing a finite geometric progression. In mathematics, the difference of two squares refers to the identity a2 âˆ’ b2 = (a + b)(a âˆ’ b) from elementary algebra. ... In mathematics, a geometric progression (also known as a geometric sequence, and, inaccurately, as a geometric series; see below) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

Infinite Series

The radical or root has the infinite series:

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

with $|x|<1$ .

Finding all of the roots

All the solutions of xⁿ = a are given by:

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

for $k=0,1,2,ldots,n-1$ .

Solving polynomials

It was once conjectured that all roots of polynomials could be expressed in terms of radicals and elementary operations. That this is not true in general is the assertion of the Abel-Ruffini theorem. For example, the solutions of the equation In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... The Abel-Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. ...

$x^5=x+1$

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 f. ...

Contents

Working with surds

Infinite Series

Finding all of the roots

Solving polynomials

See also