A binary operation ♦ is defined on the set R, of real numbers by \(a ♦ b = \frac{ab}{4}\). Find the value of \(\sqrt{2} ♦ \sqrt{6}\).

A binary operation ♦ is defined on the set R, of real numbers by \(a ♦ b = \frac{ab}{4}\). Find the value of \(\sqrt{2} ♦ \sqrt{6}\).

Answer Details

The binary operation ♦ is defined as \(a ♦ b = \frac{ab}{4}\). So, to find the value of \(\sqrt{2} ♦ \sqrt{6}\), we just need to substitute \(\sqrt{2}\) for \(a\) and \(\sqrt{6}\) for \(b\), and then simplify the expression. \[\sqrt{2} ♦ \sqrt{6} = \frac{\sqrt{2}\cdot\sqrt{6}}{4} = \frac{\sqrt{12}}{4} = \frac{\sqrt{4}\cdot\sqrt{3}}{4} = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}\] Therefore, the value of \(\sqrt{2} ♦ \sqrt{6}\) is \(\frac{\sqrt{3}}{2}\).