The three rationalisation techniques

Depending on the form of the denominator, one of the three techniques below is applied.

Property — Three techniques

1. Single radical: multiply by the complementary power. $\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \qquad \frac{3}{\sqrt[3]{2}} = \frac{3\sqrt[3]{4}}{2}$

2. Binomial with square roots: multiply by the conjugate $(a-b)$: $\frac{1}{\sqrt{2}+1} = \frac{\sqrt{2}-1}{(\sqrt{2}+1)(\sqrt{2}-1)} = \frac{\sqrt{2}-1}{2-1} = \sqrt{2}-1$

3. Binomial with cube roots: use $a^3\pm b^3 = (a\pm b)(a^2\mp ab+b^2)$: $\frac{1}{\sqrt[3]{2}+1} = \frac{\sqrt[3]{4}-\sqrt[3]{2}+1}{(\sqrt[3]{2}+1)(\sqrt[3]{4}-\sqrt[3]{2}+1)} = \frac{\sqrt[3]{4}-\sqrt[3]{2}+1}{2+1} = \frac{\sqrt[3]{4}-\sqrt[3]{2}+1}{3}$

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