<p>\(\dfrac{1}{\sqrt{2}+1} + \dfrac{1}{\sqrt{2}-1} - 2\sqrt{2}\) is equal to:</p>
Step-by-Step Solution
Key Concept: Rationalise each fraction: 1/(\sqrt{2}+1) = \sqrt{2}-1, 1/(\sqrt{2}-1) = \sqrt{2}+1. Their sum = 2\sqrt{2.} Subtract 2\sqrt{2} to get 0.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. $\dfrac{1}{\sqrt{2}+1} = \dfrac{\sqrt{2}-1}{(\sqrt{2})^2-1^2} = \sqrt{2}-1$. Similarly $\dfrac{1}{\sqrt{2}-1} = \sqrt{2}+1$. Sum = $2\sqrt{2}$. Then $2\sqrt{2}-2\sqrt{2}=0$. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: 1