Basic Mathematics & Logarithm
Surds
Grade None
Question:
<p>The square root of \(5 + 2\sqrt{6}\) is:</p>
<p>\sqrt{3} + \sqrt{2}</p>
<p>\sqrt{3} - \sqrt{2}</p>
<p>\sqrt{2} - \sqrt{3}</p>
<p>\sqrt{3} + \sqrt{2}</p>
Step-by-Step Solution
Key Concept: Write 5 + 2\sqrt{6} = 3 + 2\sqrt{6} + 2 = (\sqrt{3} + \sqrt{2})^2. Hence \sqrt{5+2\sqrt{6}} = \sqrt{3}+\sqrt{2.}
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. $5+2\sqrt{6} = 3+2\sqrt{3}\cdot\sqrt{2}+2 = (\sqrt{3}+\sqrt{2})^2$. Taking positive square root: $\sqrt{3}+\sqrt{2}$. 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