Question

(√3+1)(4√3+1)(8√3+1)(16√3+1)

Answer 1

[tex]\left(\sqrt{3}+1\right)\left(4\sqrt{3}+1\right)\left(8\sqrt{3}+1\right)\left(16\sqrt{3}+1\right) \\ \\ \left(\sqrt{3}+1\right)\left(4\sqrt{3}+1\right)=5\sqrt{3}+13 \\ \\ \:\left(5\sqrt{3}+13\right)\left(8\sqrt{3}+1\right)=109\sqrt{3}+133 \\ \\ \left(109\sqrt{3}+133\right)\left(16\sqrt{3}+1\right) \\ \\ =109\sqrt{3}\cdot \:16\sqrt{3}+109\sqrt{3}\cdot \:1+133\cdot \:16\sqrt{3}+133\cdot \:1 \\ \\ 109\sqrt{3}\cdot \:16\sqrt{3}+109\sqrt{3}\cdot \:1+133\cdot \:16\sqrt{3}+133\cdot \:1 \\ \\ [/tex]$109\sqrt{3}\cdot \:1=109\sqrt{3} \\ \\ 133\cdot \:16\sqrt{3}=2128\sqrt{3} \\ \\ 133\cdot \:1=133 \\ \\ =5232+109\sqrt{3}+2128\sqrt{3}+133 \\ \\ =5232+2237\sqrt{3}+133 \\ \\ =2237\sqrt{3}+5365$

Answer 2

(√3+1)(4√3+1)(8√3+1)(16√3+1) =
[tex](4\cdot3+ \sqrt{3}+4 \sqrt{3}+1)(16\cdot8\cdot3+8 \sqrt{3}+16 \sqrt{3}+1)= \\ \\ =(13+5 \sqrt{3})(384+24 \sqrt{3}+1) \\ \\ =(13+5 \sqrt{3})(385+24 \sqrt{3}) \\ \\ =385\cdot3+13\cdot24 \sqrt{3}+5\cdot383 \sqrt{3} +5\cdot24\cdot3 \\ \\ =5005+312 \sqrt{3}+195 \sqrt{3}+360 \\ \\ =5365+2227 \sqrt{3} [/tex]