## Thursday, April 14, 2016

### Simplify [MATH]\tiny\frac{(2^{2^0}+3^{2^0})(2^{2^1}+3^{2^1})(2^{2^2}+3^{2^2})\cdots(2^{2^{10}}+3^{2^{10}})+2^{2048}}{3^{2048}}[/MATH].

Simplify [MATH]\frac{(2^{2^0}+3^{2^0})(2^{2^1}+3^{2^1})(2^{2^2}+3^{2^2})\cdots(2^{2^{10}}+3^{2^{10}})+2^{2048}}{3^{2048}}[/MATH].

My solution:

[MATH]\frac{(2^{2^0}+3^{2^0})(2^{2^1}+3^{2^1})(2^{2^2}+3^{2^2})\cdots(2^{2^{10}}+3^{2^{10}})+2^{2048}}{3^{2048}}[/MATH]

[MATH]=\frac{(2+3)(2^{2}+3^{2})(2^{4}+3^{4})\cdots(2^{1024}+3^{1024})+2^{2048}}{3^{2048}}[/MATH]

[MATH]=\frac{(3+2)(3^{2}+2^{2})(3^{4}+2^{4})\cdots(3^{1024}+2^{1024})+2^{2048}}{3^{2048}}[/MATH]

[MATH]=\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})}[/MATH]

[MATH]=\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})}[/MATH]

[MATH]=\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})}[/MATH]

[MATH]=\frac{(3^{8}-2^{8})\cdots(3^{1024}+2^{1024})+2^{2048}}{\color{blue}(3-2)\color{black}(3^{2048})}[/MATH]

[MATH]=\frac{\underbrace{(3^{1024}-2^{1024})(3^{1024}+2^{1024})}+2^{2048}}{\color{blue}(3-2)\color{black}(3^{2048})}[/MATH]

[MATH]=\frac{(3^{2048}-2^{2048})+2^{2048}}{\color{blue}(1)\color{black}(3^{2048})}[/MATH]

[MATH]=\frac{(3^{2048}}{(3^{2048})}[/MATH]

[MATH]=1[/MATH]