What rationalisation is

Having a root in the denominator is “awkward”: rationalisation consists in rewriting the fraction so that the denominator becomes a rational number.

Definition — To rationalise

To rationalise the denominator means to rewrite a fraction in an equivalent form in which the denominator contains no roots.

The trick is always the same: multiply numerator and denominator by one and the same quantity, chosen so that the roots vanish from the denominator. The following table summarises “what to multiply by” depending on the form of the denominator.

In brief — What do I multiply by?

Denominator	Multiply by…	Why it works
$\sqrt{a}$	$\dfrac{\sqrt{a}}{\sqrt{a}}$	$\sqrt{a}\cdot\sqrt{a}=a$
$\sqrt[3]{a}$	$\dfrac{\sqrt[3]{a^2}}{\sqrt[3]{a^2}}$	$\sqrt[3]{a}\cdot\sqrt[3]{a^2}=a$
$\sqrt{a}+\sqrt{b}$	$\dfrac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}}$	$(\bigcirc+\square)(\bigcirc-\square)=\bigcirc^2-\square^2$
$\sqrt[3]{a}+\sqrt[3]{b}$	$\dfrac{\sqrt[3]{a^2}-\sqrt[3]{ab}+\sqrt[3]{b^2}}{\cdots}$	$\bigcirc^3+\square^3 = (\bigcirc+\square)(\bigcirc^2-\bigcirc\square+\square^2)$

Links

Topics: Radicals
Concepts: Conjugate · Rationalisation
Methods: Radical rationalisation
Skills: Simplifying