Simplify the Expression: (√2 × √9 × √2)/(√3 × √4) Radical Fraction

Let's proceed to simplify the expression:

• First, evaluate the numerator: $\sqrt{2} \cdot \sqrt{9} \cdot \sqrt{2}$. Using the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, we simplify it: $\sqrt{2 \cdot 9 \cdot 2} = \sqrt{36}$.
• $\sqrt{36}$ simplifies to 6, as 36 is a perfect square.
• Next, evaluate the denominator $\sqrt{3} \cdot \sqrt{4}$:
• $\sqrt{3} \cdot \sqrt{4}$ also applies the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, simplifying to $\sqrt{12}$.
• Since 12 is $4 \times 3$, and $\sqrt{4} = 2$, $\sqrt{12}$ simplifies to $2\sqrt{3}$.
• Now, the original expression becomes $\frac{6}{2\sqrt{3}}$.
• Simplify $\frac{6}{2}$ to get $\frac{6}{2} = 3$.
• The entire expression now is $\frac{3}{\sqrt{3}}$.
• To rationalize the expression $\frac{3}{\sqrt{3}}$, multiply both the numerator and the denominator by $\sqrt{3}$:
• This becomes $\frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{3\sqrt{3}}{3} = \sqrt{3}$

Therefore, the solution to the problem is $\sqrt{3}$.