Rationalisation (mathematics)

In elementary algebra, root rationalisation is a process by which radicals in the denominator of an algebraic fraction are eliminated.
If the denominator is a monomial in some radical, say formula_1 with , rationalisation consists of multiplying the numerator and the denominator by formula_2 and replacing formula_3 by (this is allowed, as, by definition, a th root of is a number that has as its th power). If , one writes with (Euclidean division), and formula_4 then one proceeds as above by multiplying by formula_5
If the denominator is linear in some square root, say formula_6 rationalisation consists of multiplying the numerator and the denominator by formula_7 and expanding the product in the denominator.
This technique may be extended to any algebraic denominator, by multiplying the numerator and the denominator by all algebraic conjugates of the denominator, and expanding the new denominator into the norm of the old denominator. 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.
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:
To rationalise this kind of expression, bring in the factor formula_9:
The square root disappears from the denominator, because formula_11 by definition of a square root:
which is the result of the rationalisation.
Example 2:
To rationalise this radical, bring in the factor formula_14:
The cube root disappears from the denominator, because it is cubed; so
which is the result of the rationalisation.
Dealing with more square roots.
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 formula_23.
Now, we can proceed to remove the square roots in the denominator:
Example 2:
This process also works with complex numbers with formula_26
The fraction must be multiplied by a quotient containing formula_28.
Generalizations.
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: