Textbook Question
Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (1 + √3) / (3√5 + 2√3)
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0. Review of Algebra
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 that needs rationalization: \( \frac{\sqrt{7}}{5 - \sqrt{6}} \).
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 5 - \sqrt{6} \) is \( 5 + \sqrt{6} \).
Perform the multiplication: \( \frac{\sqrt{7}}{5 - \sqrt{6}} \times \frac{5 + \sqrt{6}}{5 + \sqrt{6}} = \frac{\sqrt{7}(5 + \sqrt{6})}{(5 - \sqrt{6})(5 + \sqrt{6})} \).
Simplify the denominator using the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\). Here, \(a = 5\) and \(b = \sqrt{6}\), so the denominator becomes \(5^2 - (\sqrt{6})^2 = 25 - 6 = 19\).
Simplify the numerator: \( \sqrt{7}(5 + \sqrt{6}) = 5\sqrt{7} + \sqrt{42} \). Thus, the expression becomes \( \frac{5\sqrt{7} + \sqrt{42}}{19} \).
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