\(2+\sqrt{17-4.\sqrt{9+4.\sqrt{5}}}\)
\(=2+\sqrt{17-4.\sqrt{4+4.\sqrt{5}+5}}\)
\(=2+\sqrt{17-4.\sqrt{2^2+2.2.\sqrt{5}+\sqrt{5}^2}}\)
\(=2+\sqrt{17-4.\sqrt{\left(2+\sqrt{5}\right)^2}}\)
\(=2+\sqrt{17-4.\left(2+\sqrt{5}\right)}\)
\(=2+\sqrt{17-8+4.\sqrt{5}}\)
\(=2+\sqrt{9+4.\sqrt{5}}\)
\(=2+\sqrt{4+4.\sqrt{5}+5}\)
\(=2+\sqrt{2^2+2.2.\sqrt{5}+\sqrt{5}^2}\)
\(=2+\sqrt{\left(2+\sqrt{5}\right)^2}\)
\(=2+2+\sqrt{5}\)
'\(=4+\sqrt{5}\)
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