Rationalisation (mathematics)

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In elementary algebra root rationalisation is a process by surds in the denominator of a fraction are eliminated.

These surds may be monomials or binomials involving square roots, in simple examples. There are wide extensions to the technique.

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[edit] Rationalization of a monomial square root

For the fundamental technique, the numerator and denominator must be multiplied, but by the same factor.

Example:

\frac{10}{\sqrt{5}}

To rationalize this kind of monomial, bring in the factor \sqrt{5}:

\frac{10}{\sqrt{5}} = \frac{10}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{{10\sqrt{5}}}{\sqrt{5}^2}

The square root disappears from the denominator, because it is squared:

\frac{{10\sqrt{5}}}{\sqrt{5}^2} = \frac{10\sqrt{5}}{5}

This gives the result, after simplification:

\frac{{10\sqrt{5}}}{{5}} = 2\sqrt{5}

[edit] Dealing with more square roots

For a denominator that is:

\sqrt{2}+\sqrt{3}

Rationalisation can be achieved by multiplying by:

\sqrt{2}-\sqrt{3}

and applying the difference of two squares identity, which here will yield −1. To get this result, the entire fraction should be multiplied by

\frac{{{\sqrt{2}-\sqrt{3}}}}{\sqrt{2}-\sqrt{3}} = 1

This technique works much more generally. It can easily be adapted to remove one square root at a time, i.e. to rationalise

x +\sqrt{y}

by multiplication by

x -\sqrt{y}.

Example:

\frac{{3}}{\sqrt{3}+\sqrt{5}}

The fraction must be multiplied by a quotient containing {\sqrt{3}-\sqrt{5}}.

\frac{{3}}{\sqrt{3}+\sqrt{5}} · \frac{{{\sqrt{3}-\sqrt{5}}}}{\sqrt{3}-\sqrt{5}} = \frac{{3({\sqrt{3}-\sqrt{5}}) }}{\sqrt{3^2}-\sqrt{5^2}}

Now, we can proceed to remove the square roots in the denominator:

\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}}

[edit] Generalisations

Rationalisation can be extended to all algebraic numbers and algebraic functions (as an application of norm forms). For example, to rationalise a cube root, two linear factors involving cube roots of unity should be used, or equivalently a quadratic factor.

[edit] References

es:Racionalización de radicales

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. 189-199.
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