Evaluate √(19 - 6√10) + 3
√10
We need to express \(19 - 6\sqrt{10}\) in the form \(a + b - 2\sqrt{ab}\), which is \((\sqrt{a} - \sqrt{b})^2\).
We need \(a + b = 19\) and \(2\sqrt{ab} = 6\sqrt{10} \Rightarrow ab = 90\).
Solving \(a+b=19\) and \(ab=90\), the values are \(a = 10\) and \(b = 9\).
So, \(19 - 6\sqrt{10} = (\sqrt{10} - \sqrt{9})^2 = (\sqrt{10} - 3)^2\).
Since \(\sqrt{10} \approx 3.16 > 3\), \(\sqrt{19 - 6\sqrt{10}} = \sqrt{10} - 3\).
Therefore, \(\sqrt{19-6\sqrt{10}} + 3 = (\sqrt{10} - 3) + 3 = \sqrt{10}\).
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