If $\sqrt{6} = 2.449$, then the value of $\frac{3\sqrt{2}}{2\sqrt{3}}$ is

### Concept & Formula Simplify surds and fractions by rationalizing the denominator or by cancelling common radical factors between the numerator and the denominator. $$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $$ ### Step-by-Step Solution Given: $\sqrt{6} = 2.449$ We need to evaluate the expression: $\frac{3\sqrt{2}}{2\sqrt{3}}$ Let's simplify the fraction by splitting the integers from the radicals: $\frac{3}{2} \times \frac{\sqrt{2}}{\sqrt{3}}$ To eliminate the square root in the denominator, rationalize it by multiplying the top and bottom by $\sqrt{3}$: $\frac{3 \times \sqrt{2} \times \sqrt{3}}{2 \times \sqrt{3} \times \sqrt{3}}$ Combine the radicals in the numerator ($\sqrt{2} \times \sqrt{3} = \sqrt{6}$) and simplify the denominator ($\sqrt{3} \times \sqrt{3} = 3$): $\frac{3 \times \sqrt{6}}{2 \times 3}$ The $3$ in the numerator and the $3$ in the denominator cancel each other out: $\frac{\sqrt{6}}{2}$ Substitute the given value for $\sqrt{6}$: $\frac{2.449}{2}$ Perform the division: $2.449 \div 2 = 1.2245$ ### Exam Strategy & Shortcut Recognize that $3$ is simply $(\sqrt{3})^2$. So, $\frac{3}{\sqrt{3}}$ instantly simplifies to $\sqrt{3}$. The expression becomes $\frac{\sqrt{3} \times \sqrt{2}}{2}$, which is $\frac{\sqrt{6}}{2}$. Divide $2.449$ in half to get $1.2245$. This takes fewer than ten seconds. ### Common Pitfall Substituting approximate values for $\sqrt{2}$ ($1.414$) and $\sqrt{3}$ ($1.732$) directly into the initial expression creates a messy calculation ($3 \times 1.414 / 2 \times 1.732$) and amplifies rounding errors. ### Final Answer Therefore, the correct answer is **1.2245**.