Radius Ratio #4

A circle of radius t is tangent to hypotenuse, incircle and one leg of an isosceles right triangle with inradius $r = 1 + sin\frac{\pi}{8}$ .

If the value of $r \times t$ can be written as $\displaystyle \dfrac{a + \sqrt{a}}{b}$

Where 'a' is square free & a,b are integers, find the value of $a^{b} + b^{a}$ .

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