Lesson 4: Rationalizing Denominators

Property

The process of eliminating a radical from the denominator of a fraction is called rationalizing the denominator. This is done by multiplying the numerator and the denominator by the same root that appeared in the denominator originally. It is often best to simplify any radicals in the expression before attempting to rationalize.

Examples

• To rationalize $\frac{7}{\sqrt{3}}$, multiply the numerator and denominator by $\sqrt{3}$: $\frac{7 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{7\sqrt{3}}{3}$.

• To rationalize $\frac{6}{\sqrt{12}}$, first simplify the denominator: $\frac{6}{2\sqrt{3}} = \frac{3}{\sqrt{3}}$. Now rationalize: $\frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{3\sqrt{3}}{3} = \sqrt{3}$.

Property

Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer. To rationalize a denominator with a higher index radical, we multiply the numerator and denominator by a radical that would give us a radicand that is a perfect power of the index.

Examples

• To simplify $\dfrac{7}{\sqrt{5}}$, multiply by $\dfrac{\sqrt{5}}{\sqrt{5}}$ to get $\dfrac{7\sqrt{5}}{(\sqrt{5})^2} = \dfrac{7\sqrt{5}}{5}$.
• To simplify $\dfrac{2}{\sqrt[3]{9}}$, rewrite as $\dfrac{2}{\sqrt[3]{3^2}}$. Multiply by $\dfrac{\sqrt[3]{3}}{\sqrt[3]{3}}$ to get $\dfrac{2\sqrt[3]{3}}{\sqrt[3]{3^3}} = \dfrac{2\sqrt[3]{3}}{3}$.
• To simplify $\dfrac{5}{\sqrt{8x}}$, first simplify to $\dfrac{5}{2\sqrt{2x}}$. Then multiply by $\dfrac{\sqrt{2x}}{\sqrt{2x}}$ to get $\dfrac{5\sqrt{2x}}{2(2x)} = \dfrac{5\sqrt{2x}}{4x}$.

Explanation

Radicals in the denominator are considered un-simplified. To fix this, we multiply the numerator and denominator by a value that makes the denominator's radicand a perfect power of the index, which removes the radical and makes the denominator rational.