## Lesson 7: Rationalizing the Denominator In mathematics, a radical expression is not considered fully simplified if the denominator contains a square root. Rationalizing the denominator means rewriting the expression so there are **no radicals in the denominator**. This is often required in electrical calculations where impedance or voltage ratios are written in fractional form and need to be cleaned up for reporting or simplification. ### Why Rationalize? - Avoids dividing by an irrational number - Makes results easier to compare and interpret - Required by many standards for presenting final answers ### Basic Rule To remove a square root from the denominator: $ \frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a} $ You **multiply the numerator and denominator** by the radical that is in the denominator. ### Example 1: Rationalize $\frac{1}{\sqrt{2}}$ Multiply top and bottom by $\sqrt{2}$: $ \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} $ Now there is no radical in the denominator. ### Example 2: Rationalize $\frac{5}{\sqrt{3}}$ $ \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3} $ ### Example 3: With Variables $ \frac{V}{\sqrt{R}} \quad \Rightarrow \quad \frac{V\sqrt{R}}{R} $ Useful when rearranging formulas involving square roots. ### Application in Electrical Formulas Suppose you are solving for RMS voltage: $ V_{\text{RMS}} = \frac{V_{\text{peak}}}{\sqrt{2}} $ You might want to rationalize for clearer presentation: $ V_{\text{RMS}} = \frac{V_{\text{peak}}\sqrt{2}}{2} $ This version is mathematically equivalent, but avoids a radical in the denominator. ### Summary - Rationalizing removes radicals from denominators - Multiply numerator and denominator by the radical - Helps with clear reporting in technical fields like electrical testing - Common in RMS, impedance, and power formulas