If \(x = 11 + 2\sqrt{30}\), then what is the value of \(x^{1/2} + x^{-1/2}\)?
\(2\sqrt{6}\)
\(11 + 2\sqrt{30} = 6 + 5 + 2\sqrt{6}\sqrt{5} = (\sqrt{6}+\sqrt{5})^2\), so \(x^{1/2} = \sqrt{6}+\sqrt{5}\). Then \(x^{-1/2} = \dfrac{1}{\sqrt{6}+\sqrt{5}} = \sqrt{6}-\sqrt{5}\) (rationalising). Therefore \(x^{1/2}+x^{-1/2} = (\sqrt{6}+\sqrt{5}) + (\sqrt{6}-\sqrt{5}) = 2\sqrt{6}\).
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