In elementary algebra, root rationalisation (or rationalization) is a process by which radicals in the denominator of an algebraic fraction are eliminated.
with k < n, rationalisation consists of multiplying the numerator and the denominator by
If k ≥ n, one writes k = qn + r with 0 ≤ r < n (Euclidean division), and
If the denominator is linear in some square root, say
rationalisation consists of multiplying the numerator and the denominator by
However, except in special cases, the resulting fractions may have huge numerators and denominators, and, therefore, the technique is generally used only in the above elementary cases.
For the fundamental technique, the numerator and denominator must be multiplied by the same factor.
Example 1: To rationalise this kind of expression, bring in the factor
: The square root disappears from the denominator, because
by definition of a square root: which is the result of the rationalisation.
Example 2: To rationalise this radical, bring in the factor
For a denominator that is: Rationalisation can be achieved by multiplying by the conjugate: and applying the difference of two squares identity, which here will yield −1.
To get this result, the entire fraction should be multiplied by This technique works much more generally.
It can easily be adapted to remove one square root at a time, i.e. to rationalise by multiplication by Example: The fraction must be multiplied by a quotient containing
Now, we can proceed to remove the square roots in the denominator: Example 2: This process also works with complex numbers with
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.
This material is carried in classic algebra texts.