Certainly! Let's solve for the following quotient:
[tex]\[
\frac{3 \sqrt{8}}{4 \sqrt{6}}
\][/tex]
Follow these steps:
1. Simplify the square roots inside the fraction:
[tex]\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2 \sqrt{2}
\][/tex]
[tex]\[
\sqrt{6} = \sqrt{2 \cdot 3} = \sqrt{2} \cdot \sqrt{3} = \sqrt{6}
\][/tex]
2. Substitute these simplified forms back into the fraction:
[tex]\[
\frac{3 \sqrt{8}}{4 \sqrt{6}} = \frac{3 \cdot 2 \sqrt{2}}{4 \sqrt{6}} = \frac{6 \sqrt{2}}{4 \sqrt{6}}
\][/tex]
3. Simplify the fraction by dividing the numerical coefficients:
[tex]\[
\frac{6 \sqrt{2}}{4 \sqrt{6}} = \frac{6}{4} \cdot \frac{\sqrt{2}}{\sqrt{6}} = \frac{3}{2} \cdot \frac{\sqrt{2}}{\sqrt{6}}
\][/tex]
4. Simplify the square roots:
[tex]\[
\frac{\sqrt{2}}{\sqrt{6}} = \frac{\sqrt{2}}{\sqrt{2 \cdot 3}} = \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{3}} = \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\][/tex]
5. Substitute this back into the simplified fraction:
[tex]\[
\frac{3}{2} \cdot \frac{\sqrt{2}}{\sqrt{6}} = \frac{3}{2} \cdot \frac{\sqrt{3}}{3} = \frac{1}{2} \sqrt{3} = \frac{\sqrt{3}}{2}
\][/tex]
So, the simplified form of the quotient [tex]\(\frac{3 \sqrt{8}}{4 \sqrt{6}}\)[/tex] is:
[tex]\[
\boxed{\frac{\sqrt{3}}{2}}
\][/tex]
