- Rationalize radical denominators
rationalize a denominator
To rationalize a denominator means to eliminate radicals (square roots, cube roots, etc.) from the denominator of a fraction.
Rationalize the denominator.
- [latex]\frac{3}{\sqrt{5}}[/latex].
-
[latex]\frac{2}{3 + \sqrt{2}}[/latex].
- [latex]\frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}[/latex] Result: [latex]\frac{3\sqrt{5}}{5}[/latex] Simplify Multiply by the conjugate of the denominator, [latex]3 - \sqrt{2}[/latex]: [latex]\frac{2}{3 + \sqrt{2}} \times \frac{3 - \sqrt{2}}{3 - \sqrt{2}} = \frac{2(3 - \sqrt{2})}{(3 + \sqrt{2})(3 - \sqrt{2})}[/latex] [latex]= \frac{6 - 2\sqrt{2}}{9 - 2} = \frac{6 - 2\sqrt{2}}{7}[/latex] Result: [latex]\frac{6 - 2\sqrt{2}}{7}[/latex]
When there’s more than one term in the denominator, use its conjugate (change the sign between terms).