The value of \(\frac{{2\sqrt {10} }}{{\sqrt 5 \; + \;\sqrt 2 - \sqrt 7 }} - \sqrt {\frac{{\sqrt 5 - 2}}{{\sqrt 5 \; + \;2}}} - \frac{3}{{\sqrt 7 - 2}}\;\) is:
√2
Term 1: Multiply numerator and denominator by \((\sqrt5+\sqrt2)+\sqrt7\). Denominator becomes \((\sqrt5+\sqrt2)^2-7=7+2\sqrt{10}-7=2\sqrt{10}\), so Term 1 \(=\sqrt5+\sqrt2+\sqrt7\).
Term 2: Rationalising, \(\frac{\sqrt5-2}{\sqrt5+2}=\frac{(\sqrt5-2)^2}{1}\), so \(\sqrt{\frac{\sqrt5-2}{\sqrt5+2}}=\sqrt5-2\).
Term 3: \(\frac{3}{\sqrt7-2}=\frac{3(\sqrt7+2)}{7-4}=\sqrt7+2\).
Combine:
\[(\sqrt5+\sqrt2+\sqrt7)-(\sqrt5-2)-(\sqrt7+2)=\mathbf{\sqrt2}\]
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