If \(y=5+2\sqrt{6}\), what is the value of \(y+\frac{1}{y}\)?

Correct answer: 10

The conjugate of \(5+2\sqrt{6}\) is \(5-2\sqrt{6}\). Since \((5+2\sqrt{6})(5-2\sqrt{6})=25-24=1\), we get \(\frac{1}{5+2\sqrt{6}}=5-2\sqrt{6}\). Therefore, \(y+\frac{1}{y}=(5+2\sqrt{6})+(5-2\sqrt{6})=10\). Option A considers only the integer part of \(y\), while option C may result from incorrectly combining the radical terms. Exam tip: use the conjugate to rationalize the denominator in such expressions.