Simplify: \(\dfrac{3 + \sqrt{6}}{5\sqrt{3} - 2\sqrt{12} - \sqrt{32} + \sqrt{50}}\)
\(\sqrt{3}\)
Simplify each surd in the denominator: \(2\sqrt{12} = 2\cdot 2\sqrt{3} = 4\sqrt{3}\), \(\sqrt{32} = 4\sqrt{2}\) and \(\sqrt{50} = 5\sqrt{2}\).
So the denominator becomes \(5\sqrt{3} - 4\sqrt{3} - 4\sqrt{2} + 5\sqrt{2} = \sqrt{3} + \sqrt{2}\).
The expression is now \(\dfrac{3 + \sqrt{6}}{\sqrt{3} + \sqrt{2}}\). Rationalise by multiplying numerator and denominator by \(\sqrt{3} - \sqrt{2}\).
Denominator: \((\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) = 3 - 2 = 1\).
Numerator: \((3 + \sqrt{6})(\sqrt{3} - \sqrt{2}) = 3\sqrt{3} - 3\sqrt{2} + \sqrt{18} - \sqrt{12}\).
Since \(\sqrt{18} = 3\sqrt{2}\) and \(\sqrt{12} = 2\sqrt{3}\), the numerator is \(3\sqrt{3} - 3\sqrt{2} + 3\sqrt{2} - 2\sqrt{3} = \sqrt{3}\).
Therefore the value is \(\dfrac{\sqrt{3}}{1} = \sqrt{3}\).
Hence, the simplified value is \(\sqrt{3}\).
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