Simplify (220+320)(221+321)(222+322)⋯(2210+3210)+2204832048.
My solution:
(220+320)(221+321)(222+322)⋯(2210+3210)+2204832048
=(2+3)(22+32)(24+34)⋯(21024+31024)+2204832048
=(3+2)(32+22)(34+24)⋯(31024+21024)+2204832048
\displaystyle =\frac{\underbrace{(3+2)\color{blue}(3-2)}\color{black}(3^{2}+2^{2})(3^{4}+2^{4})\cdots(3^{1024}+2^{1024})+2^{2048}}{\color{blue}(3-2)\color{black}(3^{2048})}
\displaystyle =\frac{\underbrace{(3^{2}-2^{2})(3^{2}+2^{2})}(3^{4}+2^{4})\cdots(3^{1024}+2^{1024})+2^{2048}}{\color{blue}(3-2)\color{black}(3^{2048})}
\displaystyle =\frac{\underbrace{(3^{4}-2^{4})(3^{4}+2^{4})}\cdots(3^{1024}+2^{1024})+2^{2048}}{\color{blue}(3-2)\color{black}(3^{2048})}
\displaystyle =\frac{(3^{8}-2^{8})\cdots(3^{1024}+2^{1024})+2^{2048}}{\color{blue}(3-2)\color{black}(3^{2048})}
\displaystyle =\frac{\underbrace{(3^{1024}-2^{1024})(3^{1024}+2^{1024})}+2^{2048}}{\color{blue}(3-2)\color{black}(3^{2048})}
\displaystyle =\frac{(3^{2048}-2^{2048})+2^{2048}}{\color{blue}(1)\color{black}(3^{2048})}
\displaystyle =\frac{(3^{2048}}{(3^{2048})}
\displaystyle =1
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