To simplify the expression \[ \frac{(\sqrt{12} - \sqrt{3})}{(\sqrt{12} + \sqrt{3})} \] we will follow a systematic approach. ### Step 1: Simplify the Square Roots First, let's simplify \(\sqrt{12}\): \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \] Now, we can substitute this back into our expression: \[ \frac{(2\sqrt{3} - \sqrt{3})}{(2\sqrt{3} + \sqrt{3})} \] ### Step 2: Combine Like Terms Next, we simplify the numerator and the denominator: **Numerator:** \[ 2\sqrt{3} - \sqrt{3} = (2 - 1)\sqrt{3} = \sqrt{3} \] **Denominator:** \[ 2\sqrt{3} + \sqrt{3} = (2 + 1)\sqrt{3} = 3\sqrt{3} \] Now, our expression looks like this: \[ \frac{\sqrt{3}}{3\sqrt{3}} \] ### Step 3: Simplify the Fraction We can simplify this fraction by canceling \(\sqrt{3}\) in the numerator and denominator: \[ \frac{\sqrt{3}}{3\sqrt{3}} = \frac{1}{3} \] ### Final Answer Thus, the simplified form of the original expression is: \[ \frac{1}{3} \] ### Explanation of Options Now, let's analyze the options provided: - **A. zero**: This is incorrect because we have a non-zero value of \(\frac{1}{3}\). - **B. \(\frac{1}{3}\)**: This is the correct answer as we derived it through simplification. - **C. \(\frac{3}{5}\)**: This is incorrect as it does not match our simplified result. - **D. 1**: This is also incorrect because our result is \(\frac{1}{3}\), not 1. ### Common Pitfalls 1. **Not simplifying square roots**: Always simplify square roots where possible to make calculations easier. 2. **Forgetting to combine like terms**: Ensure you combine terms in the numerator and denominator before simplifying. 3. **Canceling incorrectly**: Only cancel terms that are identical in both the numerator and denominator. ### Revision Summary - Simplify square roots before substituting them back into the expression. - Combine like terms in both the numerator and denominator. - Cancel common factors carefully to avoid mistakes. - The final simplified answer for the expression \(\frac{(\sqrt{12} - \sqrt{3})}{(\sqrt{12} + \sqrt{3})}\) is \(\frac{1}{3}\).