## Simplifying Square Roots A square root is simplified when the number under the radical (the radicand) has no perfect square factors other than 1. Simplifying helps make values easier to compare and use in calculations. ### What Is a Perfect Square? A **perfect square** is a number that is the square of an integer. **Examples of perfect squares:** - $1^2 = 1$ - $2^2 = 4$ - $3^2 = 9$ - $4^2 = 16$ - $5^2 = 25$ - $6^2 = 36$ - $7^2 = 49$ - $8^2 = 64$ - $9^2 = 81$ - $10^2 = 100$ ### Basic Simplification To simplify $\sqrt{a}$: 1. Find the largest perfect square that divides evenly into the radicand. 2. Rewrite as a product under the square root: $\sqrt{a} = \sqrt{b \cdot c}$ 3. Use the rule: $\sqrt{b \cdot c} = \sqrt{b} \cdot \sqrt{c}$ 4. Simplify the square root of the perfect square. ### Example 1: Simplify $\sqrt{50}$ - Step 1: 25 is the largest perfect square that divides 50 - Step 2: $\sqrt{50} = \sqrt{25 \cdot 2}$ - Step 3: $\sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2}$ - Step 4: $\sqrt{25} = 5$, so **Answer:** $\sqrt{50} = 5\sqrt{2}$ ### Example 2: Simplify $\sqrt{72}$ - $72 = 36 \cdot 2$ - $\sqrt{72} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$ ### Prime Factorization Method Sometimes it's easier to break the number into **prime factors**, then pull out any **pairs**. **Example: Simplify $\sqrt{18}$** - Prime factorization: $18 = 2 \cdot 3 \cdot 3$ - Group the 3s: $\sqrt{2 \cdot 3^2}$ - Pull out the square: $\sqrt{3^2} = 3$ - **Answer:** $\sqrt{18} = 3\sqrt{2}$ ### When to Stop You stop simplifying when no factor inside the radical is a perfect square (except 1). **Example:** - $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$ You stop here because 3 is not a perfect square. ### Why It Matters in Electrical Work Some formulas return radical answers, especially when solving for impedance: - $Z = \sqrt{R^2 + X^2}$ You may not always calculate a clean number. Instead, you may write: - $Z = \sqrt{2500 + 3600} = \sqrt{6100}$ - You could leave it as $\sqrt{6100}$ or approximate: $\sqrt{6100} \approx 78.1\ \Omega$ Understanding simplification helps when reporting results in standard form. ### Summary - Simplify radicals by factoring out perfect squares - Use multiplication property: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ - Use prime factorization for harder problems - Only stop when no more perfect squares remain inside the radical