Math is radical!

Algebra Level 4

Consider the following equation $1+\sqrt{2}+\sqrt{2\left(2+\sqrt{2}\right)}=\sqrt{\frac{a+\sqrt{b+\sqrt{c}}}{d-\sqrt{e+\sqrt{f}}}}$ where $\large\color{#3D99F6}{a,b,c,d,e,f}$ are integers. Calculate $\large\color{#3D99F6}{a\times b\times c\times d\times e\times f}$ .