Rationalisation (mathematics)

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.

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$10sqrt{5}{sqrt{5}^2}

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

: $frac$10sqrt{5}{sqrt{5}^2} = frac{10sqrt{5{5}

This gives the result, after simplification:

: $frac$10sqrt{5}5 = 2sqrt{5}

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 = frac3({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 = frac3({sqrt{3}-sqrt{5) 3}-{5 = frac3({sqrt{3}-sqrt{5) -2

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.

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. 189-199.