Rationalizing Denominators

Rationalizing a denominator means to eliminate any irrational numbers (like square roots) from the denominator of a fraction.

$$\frac{\sqrt{3}}{3}$$

$$\frac{\sqrt{5}}{5}$$

$$\frac{2\sqrt{8}}{8} = \frac{\sqrt{8}}{4} = \frac{\sqrt{2}}{2}$$

Multiply the numerator and denominator by $$\sqrt{b}$$ to get $$\frac{a\sqrt{b}}{b}$$ .

$$\frac{5\sqrt{10}}{10} = \frac{\sqrt{10}}{2}$$

The purpose is to simplify the expression and make it easier to work with, especially in further calculations.

Multiply by the conjugate: $$\frac{3(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{3(\sqrt{2} - 1)}{1} = 3\sqrt{2} - 3$$ .

The conjugate is $$\sqrt{a} - b$$ .

Multiply by the conjugate: $$\frac{4(\sqrt{6} + 2)}{(\sqrt{6} - 2)(\sqrt{6} + 2)} = \frac{4(\sqrt{6} + 2)}{2} = 2(\sqrt{6} + 2)$$ .