Question

Arătați că:
$( \sqrt{2} + 1)( \sqrt[4]{2} + 1)( \sqrt[8]{2} + 1)( \sqrt[16]{2} + 1) = \frac{1}{ \sqrt[16]{2} - 1}$

Answer 1

$(\sqrt{2}+1)(\sqrt[4]{2}+1)(\sqrt[8]{2}+1)(\sqrt[16]{2}+1)=\dfrac{1}{\sqrt[16]{2}-1}\\(\sqrt{2}+1)(\sqrt[4]{2}+1)(\sqrt[8]{2}+1)(\sqrt[16]{2}+1)(\sqrt[16]{2}-1)=1\\(\sqrt{2}+1)(\sqrt[4]{2}+1)(\sqrt[8]{2}+1)(\sqrt[8]{2}-1)=1\\(\sqrt{2}+1)(\sqrt[4]{2}+1)(\sqrt[4]{2}-1)=1\\(\sqrt{2}+1)(\sqrt{2}-1)=1\\2-1=1\\1=1- Adevarat\\\text{Am folosit formula :} \boxed{(a+b)(a-b)=a^2-b^2}$