Let's solve this step by step. The expression given is:

$\left( \sqrt{9} \times \frac{4}{5} \div \frac{3}{1} \right) \frac{1}{\sqrt{16}}$

1. Evaluate $\sqrt{9}$ and $\sqrt{16}$:

   • $\sqrt{9} = 3$
   • $\sqrt{16} = 4$

   So, the expression becomes:

   $\left( 3 \times \frac{4}{5} \div \frac{3}{1} \right) \frac{1}{4}$

2. Perform the operations inside the parentheses:

   • First, multiply $3$ by $\frac{4}{5}$:

     $3 \times \frac{4}{5} = \frac{12}{5}$

   • Next, divide $\frac{12}{5}$ by $\frac{3}{1}$:

     $\frac{12}{5} \div \frac{3}{1} = \frac{12}{5} \times \frac{1}{3} = \frac{12 \cdot 1}{5 \cdot 3} = \frac{12}{15} = \frac{4}{5}$

   So, the expression inside the parentheses simplifies to $\frac{4}{5}$.

3. Combine the result with $\frac{1}{4}$:

   Now we have:

   $\frac{4}{5} \times \frac{1}{4} = \frac{4 \cdot 1}{5 \cdot 4} = \frac{4}{20} = \frac{1}{5}$

The final answer in simplest form is:

$\frac{1}{5}$

Would you like a deeper explanation of any of these steps, or do you have questions?

Additional Questions:

1. How would this expression change if we used different square roots in the initial terms?
2. How can we simplify fractions when multiplying and dividing within parentheses?
3. What steps would be different if the square roots were irrational numbers?
4. How do we handle expressions with both radicals and fractions in the denominator?
5. Could this solution method be applied to more complex expressions with nested parentheses?

Tip:

When dealing with mixed operations, always resolve the expressions inside parentheses first, and handle square roots and fractions carefully for accurate simplification.