rationaliser

Rationalising refers to the process of eliminating radicals (such as square roots or cube roots) from the denominator of a fraction. This technique is commonly used in mathematics, particularly in algebra and calculus, to simplify expressions and make them easier to work with. The goal is to rewrite the expression in a form that does not contain radicals in the denominator, which is often considered more conventional and cleaner.

The most frequent application of rationalising occurs when dealing with fractions containing square roots. For example, in an expression like 1/√2, the denominator is irrational. To rationalise this, the denominator is multiplied by √2, resulting in √2/2. This maintains the value of the fraction while removing the radical from the denominator. The same principle applies to more complex expressions, such as those involving binomials with square roots, such as 1/(a + √b). In such cases, the conjugate of the denominator (a - √b) is used to multiply both the numerator and the denominator, which cancels out the radical when expanded.

Rationalising is also applicable to higher-order roots, such as cube roots, though the process may involve more steps. For instance, in an expression like 1/∛3, the denominator can be rationalised by multiplying the numerator and denominator by ∛9 (since 3 × 9 = 27, and ∛27 = 3). This results in ∛9/3, which is a rationalised form.

The technique is particularly useful in simplifying expressions before performing further operations, such as integration or differentiation, where irrational denominators can complicate calculations. While rationalising does not change the value of the expression, it often makes it more manageable and aligns with standard mathematical conventions.