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Thursday, April 14, 2016

Simplify (220+320)(221+321)(222+322)(2210+3210)+2204832048.

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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