Simplify the expression: \(\sqrt{9 + 4\sqrt{5}}\)
\(\sqrt{5} + 2\)
Step 1 – express the radicand as a perfect square:
Try \((a + b)^2 = a^2 + b^2 + 2ab = 9 + 4\sqrt{5}\).
This needs \(a^2 + b^2 = 9\) and \(2ab = 4\sqrt{5}\).
Step 2 – pick a, b:
Take \(a = 2,\ b = \sqrt{5}\): then \(a^2 + b^2 = 4 + 5 = 9\) ✓ and \(2ab = 2 \cdot 2 \cdot \sqrt{5} = 4\sqrt{5}\) ✓.
Step 3 – take the square root:
\(\sqrt{9 + 4\sqrt{5}} = \sqrt{(2 + \sqrt{5})^2} = 2 + \sqrt{5} = \sqrt{5} + 2\).
Hence the answer is \(\sqrt{5} + 2\).
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