The given expression is:

$\sqrt{45 \cdot z} \cdot \sqrt{\frac{16 \cdot z}{5}}$

Let's simplify this step by step.

Step 1: Combine the square roots

We can combine the square roots into a single expression: $\sqrt{45 \cdot z} \cdot \sqrt{\frac{16 \cdot z}{5}} = \sqrt{(45 \cdot z) \cdot \left(\frac{16 \cdot z}{5}\right)}$

Step 2: Multiply inside the square root

$= \sqrt{\frac{45 \cdot z \cdot 16 \cdot z}{5}}$ $= \sqrt{\frac{720 \cdot z^2}{5}}$ Dividing 720 by 5: $= \sqrt{144 \cdot z^2}$

Step 3: Simplify the square root

$= \sqrt{144} \cdot \sqrt{z^2}$ $= 12 \cdot z$

The final simplified expression is: $12z$

Would you like a detailed explanation of any part of this solution, or do you have any questions?

Here are some related questions that you might find helpful:

1. How do you combine multiple square roots into one?
2. What properties of square roots are essential for simplifying expressions?
3. How do you handle multiplication of fractions within square roots?
4. How do you simplify expressions involving variables and exponents inside square roots?
5. What other methods can be used to simplify similar radical expressions?

Tip: When multiplying square roots, it's often easier to combine them into a single square root to simplify calculations.