Radical (mathematics)

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See radical for other uses of the term.

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 it is assumed you are talking about the principal n-th root of the number, which is denotated <math>\sqrt[n]{a}</math>. 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.

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

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Fundamental operations

Operations with radicals are given by the following formulas:

<math>

\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}, </math>

<math>

\sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m = \left(a^{\frac{1}{n}}\right)^m = a^{\frac{m}{n}}, </math>

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 <math>\sqrt[n]{a}</math> 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 exponenets still apply (even to fractional exponents), namely

For example:

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

<math>\sqrt[3]{a^5} = \sqrt[3]{aaaaa} = \sqrt[3]{a^3a^2} = a\sqrt[3]{a^2}</math>

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

For example,

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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:

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

Square roots, cube roots and so on, can all remain in surd form.

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

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

<math>(\sqrt{a}+\sqrt{b})(\sqrt{a}- \sqrt{b}) = a - b</math>,

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.

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Infinite Series

The radical or root has the infinite series:

<math>

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

with <math>\ |x|<1</math>.

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Finding all of the roots

All the solutions of xⁿ = a are given by:

<math> e^{2\pi i \frac{k}{n}} \times \sqrt[n]{a}</math>

for <math>k=0,1,2,\ldots,n-1</math>.

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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

cannot be expressed in terms of radicals.

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

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Radical (mathematics)

From Free net encyclopedia

Contents

Fundamental operations

Working with surds

Infinite Series

Finding all of the roots

Solving polynomials

See also

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