Question

what is the following quotient?
\\(\frac{3\sqrt{8}}{4\sqrt{6}}\\)
\\(\bigcirc\\) \\(\frac{12\sqrt{2}-6\sqrt{3}}{5}\\)
\\(\bigcirc\\) \\(\frac{3\sqrt{6}-4\sqrt{3}}{24}\\)
\\(\bigcirc\\) \\(\frac{\sqrt{3}}{12}\\)
\\(\bigcirc\\) \\(\frac{\sqrt{3}}{2}\\)

Explanation:

Step1: Simplify the square roots

Simplify \(\sqrt{8}\) as \(2\sqrt{2}\), so the numerator becomes \(3\times2\sqrt{2} = 6\sqrt{2}\). The expression is now \(\frac{6\sqrt{2}}{4\sqrt{6}}\).

Step2: Rationalize the denominator

Multiply numerator and denominator by \(\sqrt{6}\): \(\frac{6\sqrt{2}\times\sqrt{6}}{4\sqrt{6}\times\sqrt{6}}\).

Step3: Simplify the products

\(\sqrt{2}\times\sqrt{6}=\sqrt{12}=2\sqrt{3}\) and \(\sqrt{6}\times\sqrt{6}=6\). So we have \(\frac{6\times2\sqrt{3}}{4\times6}\).

Step4: Simplify the fraction

\(\frac{12\sqrt{3}}{24}=\frac{\sqrt{3}}{2}\).

Answer:

\(\frac{\sqrt{3}}{2}\) (the last option, which should be presented as per the multiple - choice format: D. \(\frac{\sqrt{3}}{2}\) assuming the options are labeled A, B, C, D in order)