Question : Find the mean proportion between $(6+\sqrt{8})$ and $(3-\sqrt{2})$.
Option 1: $2 \sqrt{12}$
Option 2: $\sqrt{14}$
Option 3: $(6-\sqrt{8})$
Option 4: $\sqrt{15}-7$
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Correct Answer: $\sqrt{14}$
Solution :
The mean proportion between $(6+\sqrt{8})$ and $(3-\sqrt{2})$ is:
$\sqrt{(6+\sqrt{8})(3-\sqrt{2})}$
$=\sqrt{18-6\sqrt{2}+3\sqrt{8}-\sqrt{16}}$
$=\sqrt{18-6\sqrt{2}+6\sqrt{2}-4}$
$=\sqrt{14}$
Hence, the correct answer is $\sqrt{14}$.
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