Multiple Choice
Rationalize the denominator.
0. Review of College Algebra
Rationalizing Denominators
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Rationalize the denominator and simplify the radical expression.
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Verified step by step guidance1
Identify the expression to be rationalized: \( \frac{2 - \sqrt{3}}{2 + \sqrt{3}} \). The goal is to eliminate the square root in the denominator.
Multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + \sqrt{3}\) is \(2 - \sqrt{3}\). This gives: \( \frac{(2 - \sqrt{3})(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} \).
Apply the difference of squares formula to the denominator: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 2\) and \(b = \sqrt{3}\), so the denominator becomes \(2^2 - (\sqrt{3})^2 = 4 - 3 = 1\).
Expand the numerator: \((2 - \sqrt{3})(2 - \sqrt{3}) = 2^2 - 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 - 4\sqrt{3} + 3\).
Simplify the expression: The numerator simplifies to \(4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3}\). Since the denominator is 1, the expression simplifies to \(7 - 4\sqrt{3}\).
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