Problem: The expression 2+2+12+2+12−22+\sqrt{2}+\dfrac{1}{2+\sqrt{2}}+\dfrac{1}{\sqrt{2}-2}2+2+2+21+2−21 equals:
Answer Choices:
A. 222
B. 2−22-\sqrt{2}2−2
C. 2+22+\sqrt{2}2+2
D. 222 \sqrt{2}22
E. 2/2\sqrt{2} / 22/2 Solution:
(2+2)2(2−2)+2−2+2+2(2+2)(2−2)=−4−2=2\dfrac{(2+\sqrt{2})^{2}(\sqrt{2}-2)+\sqrt{2}-2+2+\sqrt{2}}{(2+\sqrt{2})(\sqrt{2}-2)}=\dfrac{-4}{-2}=2(2+2)(2−2)(2+2)2(2−2)+2−2+2+2=−2−4=2