To rationalize the denominator of the expression \(\frac{11}{\sqrt{17} - 4}\), we can multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{17} + 4\).
Here are the steps:
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Write the expression: \[ \frac{11}{\sqrt{17} - 4} \]
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Multiply the numerator and the denominator by the conjugate of the denominator: \[ \frac{11(\sqrt{17} + 4)}{(\sqrt{17} - 4)(\sqrt{17} + 4)} \]
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Simplify the denominator using the difference of squares: \[ (\sqrt{17})^2 - (4)^2 = 17 - 16 = 1 \]
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Now, rewrite the expression: \[ \frac{11(\sqrt{17} + 4)}{1} = 11(\sqrt{17} + 4) = 11\sqrt{17} + 44 \]
Thus, the rationalized form of \(\frac{11}{\sqrt{17} - 4}\) is: \[ 11\sqrt{17} + 44 \]