To simplify the expression \(\frac{1}{\sqrt{3}+2}\) into the form \(a + b\sqrt{3}\), we will follow a systematic approach. ### Step 1: Rationalizing the Denominator The first step in simplifying this expression is to eliminate the square root from the denominator. We can do this by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{3} + 2\) is \(\sqrt{3} - 2\). So, we multiply: \[ \frac{1}{\sqrt{3}+2} \cdot \frac{\sqrt{3}-2}{\sqrt{3}-2} = \frac{\sqrt{3}-2}{(\sqrt{3}+2)(\sqrt{3}-2)} \] ### Step 2: Simplifying the Denominator Next, we need to simplify the denominator: \[ (\sqrt{3}+2)(\sqrt{3}-2) = \sqrt{3}^2 - 2^2 = 3 - 4 = -1 \] ### Step 3: Putting It All Together Now, substituting back into our expression, we have: \[ \frac{\sqrt{3}-2}{-1} = -(\sqrt{3}-2) = -\sqrt{3} + 2 \] ### Step 4: Rearranging the Expression We can rearrange this to match the form \(a + b\sqrt{3}\): \[ 2 - \sqrt{3} \] ### Conclusion Thus, the simplified form of \(\frac{1}{\sqrt{3}+2}\) is: \[ 2 - \sqrt{3} \] ### Final Answer The correct option is **A. 2 - √3**. ### Explanation of Other Options - **B. -2 - √3**: This option is incorrect because it has the wrong signs for both terms. The correct simplification yields a positive \(2\) and a negative \(\sqrt{3}\). - **C. 2 + √3**: This option is incorrect because it incorrectly adds \(\sqrt{3}\) instead of subtracting it. The correct simplification shows that \(\sqrt{3}\) is negative in the final expression. - **D. -2 + √3**: This option is incorrect as it has the wrong sign for the constant term and the square root term. The correct simplification shows a positive \(2\) and a negative \(\sqrt{3}\). ### Revision Summary - To simplify expressions with square roots in the denominator, multiply by the conjugate. - Rationalizing the denominator helps eliminate square roots and simplifies the expression. - Always rearrange the final expression to match the required form. - Check each option carefully to ensure the signs and terms match the simplified result.