Rationalisation (mathematics)

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For other uses, see Rationalization.

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

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

Rationalisation of a monomial square root and cube root[edit]

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

Example 1:

\frac{10}{\sqrt{a}}

To rationalise this kind of monomial, bring in the factor \sqrt{a}:

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

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

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

This gives the result, after simplification:

\frac{{10\sqrt{a}}}{{a}}

Example 2:

\frac{10}{\sqrt[3]{b}}

To rationalise this radical, bring in the factor \sqrt[3]{b}^2:

\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}

The cube root disappears from the denominator, because it is cubed:

\frac{{10\sqrt[3]{b}^2}}{\sqrt[3]{b}^3} = \frac{10\sqrt[3]{b}^2}{b}

This gives the result, after simplification:

\frac{{10\sqrt[3]{b}^2}}{{b}}

Dealing with more square roots[edit]

For a denominator that is:

\sqrt{2}+\sqrt{3}\,

Rationalisation can be achieved by multiplying by the Conjugate:

\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}} \cdot \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}

Generalisations[edit]

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.

See also[edit]

References[edit]

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.