## Adding and Subtracting Radicals You can only add or subtract radicals when they are **like radicals**, that is, they have the same radicand and same root. Think of radicals like variables: Just as you can add $3x + 2x = 5x$, you can add $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$. ### Like vs. Unlike Radicals - **Like radicals** have the same radicand: $\sqrt{5} + 2\sqrt{5} = 3\sqrt{5}$ - **Unlike radicals** cannot be combined directly: $\sqrt{2} + \sqrt{3}$ is already simplified and stays as is. ### Steps for Adding/Subtracting Radicals 1. **Simplify each radical** if possible. 2. **Check if they are like radicals**. 3. **Combine the coefficients**, leave the radical part unchanged. ### Example 1: Add Like Radicals $4\sqrt{3} + 5\sqrt{3} = (4 + 5)\sqrt{3} = 9\sqrt{3}$ ### Example 2: Subtract Like Radicals $7\sqrt{2} - 3\sqrt{2} = 4\sqrt{2}$ ### Example 3: Simplify First, Then Add $\sqrt{8} + \sqrt{18}$ Step 1: Simplify both: - $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$ - $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$ Step 2: Combine like radicals: - $2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$ ### Real-World Electrical Example Sometimes test data gives mixed values: - Reading 1: $2\sqrt{3}$ amps - Reading 2: $\sqrt{12}$ amps Simplify $\sqrt{12} = 2\sqrt{3}$ Now you can add: - $2\sqrt{3} + 2\sqrt{3} = 4\sqrt{3}$ amps ### What Not to Do Never try to add or subtract radicals with different radicands: - $\sqrt{5} + \sqrt{6}$ This is already simplified. You **cannot** combine these. ### Summary - Only **like radicals** can be added or subtracted - Simplify each radical before combining - Treat the radicals like terms with variables - Used often when summing square-root-based measurements or derived values