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2^{\sqrt{200}} = 2^{\sqrt{100}*\sqrt{2}} = 2^{10*\sqrt{2}} = \sqrt{2}^{2*10*\sqrt{2}} = \sqrt{2}^{5*4*\sqrt{2}} = (4*\sqrt{2})^{4*\sqrt{2}} = a^a
a=4*\sqrt{2}
a^2=32
a=4*\sqrt{2}anda^2=32.