<p>If \(x = 8 - \sqrt{60}\), then \(\sqrt{\dfrac{x}{2}} + \sqrt{\dfrac{2}{x}} =\)</p>
Step-by-Step Solution
Key Concept: Note \sqrt{60} = 2\sqrt{15.} Rationalise x = 8-2\sqrt{15} = (\sqrt{5}-\sqrt{3})^2 \times something. Then simplify the surd expression.
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. $x = 8-2\sqrt{15} = (\sqrt{5}-\sqrt{3})^2 \cdot \text{(factor)}$. Compute $\sqrt{x/2}+\sqrt{2/x}$ by rationalising to get $2\sqrt{5}$. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: B