Problem: The expression 2+2+12+2+12−22+\sqrt{2}+\dfrac{1}{2+\sqrt{2}}+\dfrac{1}{\sqrt{2}-2}2+2​+2+2​1​+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