sqrt(2*sqrt(2)+3)-sqrt(2)

Let the expression be $$x = \sqrt{2\sqrt{2}+3}-\sqrt{2}$$.

Squaring both sides: $$x^2 = (\sqrt{2\sqrt{2}+3}-\sqrt{2})^2 = (2\sqrt{2}+3) - 2\sqrt{2}\sqrt{2\sqrt{2}+3} + 2 = 2\sqrt{2}+5 - 2\sqrt{4\sqrt{2}+6}$$.

This expression does not simplify easily to a rational number or a simple radical form.