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In [[elementary algebra]], '''root rationalisation''' is a process by which [[nth root|surds]] in the [[denominator]] of an [[Fraction (mathematics)#Algebraic fractions|irrational fraction]] are eliminated.  
 
These surds may be [[monomial]]s or [[binomial]]s involving [[square root]]s, in simple examples. There are wide extensions to the technique.
 
== Rationalisation of a monomial square root and cube root ==
 
For the fundamental technique, the numerator and denominator must be multiplied by the same factor.
 
Example 1:
 
: <math>\frac{10}{\sqrt{a}}</math>
 
To rationalise this kind of [[monomial]], bring in the factor <math>\sqrt{a}</math>:
 
: <math>\frac{10}{\sqrt{a}} = \frac{10}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}} = \frac{{10\sqrt{a}}}{\sqrt{a}^2}</math>
 
The [[square root]] disappears from the denominator, because it is squared:
 
: <math>\frac{{10\sqrt{a}}}{\sqrt{a}^2} = \frac{10\sqrt{a}}{a}</math>  
 
This gives the result, after simplification:
 
: <math>\frac{{10\sqrt{a}}}{{a}}</math>
 
Example 2:
 
: <math>\frac{10}{\sqrt[3]{b}}</math>
 
To rationalise this radical, bring in the factor <math>\sqrt[3]{b}^2</math>:
 
: <math>\frac{10}{\sqrt[3]{b}} = \frac{10}{\sqrt[3]{b}} \cdot \frac{\sqrt[3]{b}^2}{\sqrt[3]{b}^2} = \frac{{10\sqrt[3]{b}^2}}{\sqrt[3]{b}^3}</math>
 
The cube root disappears from the denominator, because it is cubed:
 
<math>\frac{{10\sqrt[3]{b}^2}}{\sqrt[3]{b}^3} = \frac{10\sqrt[3]{b}^2}{b}</math>
 
This gives the result, after simplification:
 
: <math>\frac{{10\sqrt[3]{b}^2}}{{b}}</math>
 
== Dealing with more square roots ==
 
For a denominator that is:
 
:<math>\sqrt{2}+\sqrt{3}\,</math>
 
Rationalisation can be achieved by multiplying by the ''[[Conjugate (algebra)|Conjugate]]'':
 
:<math>\sqrt{2}-\sqrt{3}\,</math>
 
and applying the [[difference of two squares]] identity, which here will yield &minus;1. To get this result, the entire fraction should be multiplied by
 
:<math>\frac{ \sqrt{2}-\sqrt{3} }{\sqrt{2}-\sqrt{3}} = 1.</math>
 
This technique works much more generally. It can easily be adapted to remove one square root at a time, i.e. to rationalise
 
:<math>x +\sqrt{y}\,</math>
 
by multiplication by
 
:<math>x -\sqrt{y}</math>
 
Example:
 
:<math>\frac{3}{\sqrt{3}+\sqrt{5}}</math>
 
The fraction must be multiplied by a quotient containing <math>{\sqrt{3}-\sqrt{5}}</math>.
 
:<math>\frac{3}{\sqrt{3}+\sqrt{5}} \cdot \frac{\sqrt{3}-\sqrt{5}}{\sqrt{3}-\sqrt{5}} = \frac{3(\sqrt{3}-\sqrt{5})}{\sqrt{3}^2 - \sqrt{5}^2}</math>
 
Now, we can proceed to remove the square roots in the denominator:
 
: <math>\frac{{3(\sqrt{3}-\sqrt{5}) }}{\sqrt{3}^2 - \sqrt{5}^2} = \frac{ 3 (\sqrt{3} - \sqrt{5} ) }{ 3 - 5 } = \frac{ 3( \sqrt{3}-\sqrt{5} )  }{-2}</math>
 
==Generalisations==
 
Rationalisation can be extended to all [[algebraic number]]s and [[algebraic function]]s (as an application of [[norm form]]s). For example, to rationalise a [[cube root]], two linear factors involving [[cube roots of unity]] should be used, or equivalently a quadratic factor.
 
==See also==
*[[Conjugate (algebra)]]
*[[Sum of two squares]]{{dn|date=December 2013}}
 
==References==
This material is carried in classic algebra texts. For example:
 
*[[George Chrystal]], ''Introduction to Algebra: For the Use of Secondary Schools and Technical Colleges'' is a nineteenth-century text, first edition 1889, in print (ISBN 1402159072); a trinomial example with square roots is on p. 256, while a general theory of rationalising factors for surds is on pp. 189&ndash;199.
 
[[Category:Elementary algebra]]
[[Category:Fractions]]

Latest revision as of 17:31, 2 January 2015

Proper dіet is vital for everyone. An individual is possiƄly receiving the vitamins and minerals he reqսirements, or he isn't. You either ɦave the option to eat inadequately, or a healthy diet. The following sentences have numerous suցցestiߋns that can be used to consume bеtter and not merely by having a constrained diet of only h2o and having salаd.

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Use wаtеr to prepare vegetables as oρposed to oil. Steaming and Ьoіling hot fresh vegetaƅles are yummy and better foг you than fried types. Nonetheless, if you choose that a little bit oils should be used, then use a tiny bit of vegetable gas as ɑn alternative to butter or margarine.

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A diabetic person nutrients prepare mandatеs that you speaҡ with your doctor about аny feasible alcoholic beverages restrictions. Liquor can reduced blood sugar, which can oƄviously create prօblems fοr diabetics.

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