Multiple Choice
Rationalize the denominator and simplify the radical expression.
0. Fundamental Concepts 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: \( \frac{\sqrt{7}}{5 - \sqrt{6}} \). 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 \( 5 - \sqrt{6} \) is \( 5 + \sqrt{6} \). This gives: \( \frac{\sqrt{7} \cdot (5 + \sqrt{6})}{(5 - \sqrt{6})(5 + \sqrt{6})} \).
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 \(5^2 - (\sqrt{6})^2 = 25 - 6 = 19\).
Distribute \(\sqrt{7}\) in the numerator: \(\sqrt{7} \cdot 5 + \sqrt{7} \cdot \sqrt{6} = 5\sqrt{7} + \sqrt{42}\).
Combine the results to form the rationalized expression: \( \frac{5\sqrt{7} + \sqrt{42}}{19} \). This is the simplified form of the original expression.
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