Problem: The expression 1−11+3+11−31-\dfrac{1}{1+\sqrt{3}}+\dfrac{1}{1-\sqrt{3}}1−1+31+1−31 equals
Answer Choices:
A. 1−31-\sqrt{3}1−3
B. 111
C. −3-\sqrt{3}−3
D. 3\sqrt{3}3
E. 1+31+\sqrt{3}1+3 Solution:
1⋅(1+3)(1−3)−1⋅(1−3)+1⋅(1+3)(1+3)(1−3)=1−3\dfrac{1 \cdot(1+\sqrt{3})(1-\sqrt{3})-1 \cdot(1-\sqrt{3})+1 \cdot(1+\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})}=1-\sqrt{3}(1+3)(1−3)1⋅(1+3)(1−3)−1⋅(1−3)+1⋅(1+3)=1−3.