Question

the mean proportional between (3 + √2) and (12 - √32)

Answer 1

Dear student
$$\begin{gathered} \mathrm{Mean}\mathrm{propportional}\mathrm{between}\left(3+\sqrt{2}\right)\mathrm{and}\left(12-\sqrt{32}\right) \\ =\sqrt{\left(3+\sqrt{2}\right)\left(12-\sqrt{32}\right)} \\ =\sqrt{\left(3+\sqrt{2}\right)\left(12-4\sqrt{2}\right)} \\ =\sqrt{4\left(3+\sqrt{2}\right)\left(3-\sqrt{2}\right)} \\ =2\sqrt{\left(3+\sqrt{2}\right)\left(3-\sqrt{2}\right)} \\ =2\sqrt{3^2-{\left(\sqrt{2}\right)}^2} \\ =2\sqrt{9-2} \\ =2\sqrt{7} \end{gathered}$$
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