Radius Ratio #4

Geometry Level 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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