Question : The value of $\sqrt{9-2 \sqrt{11-6 \sqrt{2}}}$ is closest to:
Option 1: 2.7
Option 2: 2.9
Option 3: 2.4
Option 4: 2.1
Correct Answer: 2.4
Solution :
Given: $\sqrt{9-2 \sqrt{11-6 \sqrt{2}}}$
= $\sqrt{9-2 \sqrt{3^2+(\sqrt2)^2-2×3×\sqrt{2}}}$
= $\sqrt{9-2 \sqrt{(3-\sqrt2)^2}}$
= $\sqrt{9-2 (3-\sqrt2)}$
= $\sqrt{9-6+2\sqrt2}$
= $\sqrt{3+2\sqrt2}$
= $\sqrt{(\sqrt2)^2+1^2+2\sqrt2}$
= $\sqrt{(\sqrt2+1)^2}$
= $\sqrt2+1$
= 1.414 + 1
= 2.414
Hence, the correct answer is 2.4.
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