Evaluate \(\sqrt{13 + 4\sqrt{10}}\)
\(2\sqrt{2} +\sqrt{5}\)
We want to write \(13 + 4\sqrt{10}\) in the form \((\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab}\).
Comparing, \(a + b = 13\) and \(2\sqrt{ab} = 4\sqrt{10}\), so \(\sqrt{ab} = 2\sqrt{10}\) and \(ab = 40\).
Solving \(a + b = 13\) and \(ab = 40\) gives \(a = 8\) and \(b = 5\) (roots of \(t^2 - 13t + 40 = 0\)).
So \(13 + 4\sqrt{10} = (\sqrt{8} + \sqrt{5})^2\), and \(\sqrt{13 + 4\sqrt{10}} = \sqrt{8} + \sqrt{5} = 2\sqrt{2} + \sqrt{5}\).
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