Textbook Question
Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (9 - r) / (3 - √r)
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Rationalize Denominator
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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: \( \frac{2 - \sqrt{3}}{2 + \sqrt{3}} \).
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2 + \sqrt{3} \) is \( 2 - \sqrt{3} \).
Perform the multiplication: \( \frac{(2 - \sqrt{3})(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} \).
Simplify the denominator using the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 2\) and \(b = \sqrt{3}\), so the denominator becomes \(2^2 - (\sqrt{3})^2 = 4 - 3 = 1\).
Simplify the numerator by expanding \((2 - \sqrt{3})^2\) using the formula \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 2\) and \(b = \sqrt{3}\), so the numerator becomes \(4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3}\).
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