Rationalize the denominator and simplify the radical expression.
$\frac{2-\sqrt3}{2+\sqrt3}$

• A. $7-4\sqrt3$
• B. $\frac{1}{7+4\sqrt3}$
• C. $7+4\sqrt3$
• D. $\frac{1}{7-4\sqrt3}$

Answer: (A)

Verified step by step guidance

1. Identify the expression to be rationalized: \( \frac{2 - \sqrt{3}}{2 + \sqrt{3}} \). The goal is to eliminate the square root in the denominator.
2. Multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + \sqrt{3}\) is \(2 - \sqrt{3}\).
3. Perform the multiplication: \( (2 - \sqrt{3})(2 - \sqrt{3}) \) for the numerator and \( (2 + \sqrt{3})(2 - \sqrt{3}) \) for the denominator.
4. Use the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\). Apply this to the denominator: \((2)^2 - (\sqrt{3})^2 = 4 - 3 = 1\).
5. Simplify the expression: The denominator becomes 1, so the expression simplifies to \( (2 - \sqrt{3})^2 \). Expand and simplify the numerator to get the final expression.