Question

what is the following quotient?
\frac{3\sqrt{8}}{4\sqrt{6}}
options:
\frac{\sqrt{3}}{2}
\frac{12\sqrt{2}-6\sqrt{3}}{5}
\frac{\sqrt{3}}{12}
\frac{3\sqrt{6}-4\sqrt{3}}{21}

Explanation:

Step1: Simplify the numerator and denominator

First, simplify \(\sqrt{8}\) as \(2\sqrt{2}\), so the expression becomes \(\frac{3\times2\sqrt{2}}{4\sqrt{6}}=\frac{6\sqrt{2}}{4\sqrt{6}}\). Then, simplify the fraction \(\frac{6}{4}\) to \(\frac{3}{2}\), getting \(\frac{3\sqrt{2}}{2\sqrt{6}}\).

Step2: Rationalize the denominator

Multiply the numerator and denominator by \(\sqrt{6}\) to rationalize the denominator: \(\frac{3\sqrt{2}\times\sqrt{6}}{2\sqrt{6}\times\sqrt{6}}\). Using the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\), the numerator becomes \(3\sqrt{12}\) and the denominator becomes \(2\times6 = 12\).

Step3: Simplify the numerator further

Simplify \(\sqrt{12}\) as \(2\sqrt{3}\), so the numerator is \(3\times2\sqrt{3}=6\sqrt{3}\). Now the expression is \(\frac{6\sqrt{3}}{12}\).

Step4: Simplify the fraction

Simplify \(\frac{6\sqrt{3}}{12}\) by dividing numerator and denominator by 6, resulting in \(\frac{\sqrt{3}}{2}\).

Answer:

\(\frac{\sqrt{3}}{2}\) (the second option from the top, labeled \(\frac{\sqrt{3}}{2}\))