In elementary algebra, root rationalisation is a process by which radicals in the denominator of an algebraic fraction are eliminated.
Contents
- Rationalisation of a monomial square root and cube root
- Dealing with more square roots
- Generalizations
- References
If the denominator is a monomial in some radical, say
If the denominator is linear in some square root, say
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 monomial, bring in the factor
The square root disappears from the denominator, because it is squared:
This gives the result, after simplification:
Example 2:
To rationalise this radical, bring in the factor
The cube root disappears from the denominator, because it is cubed:
This gives the result, after simplification:
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
Now, we can proceed to remove the square roots in the denominator:
Example 2:
This process also works with complex numbers with
The fraction must be multiplied by a quotient containing
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.