The problem asks for the quotient of $20\sqrt{z^6} \div \sqrt{16z^7}$ in its simplest radical form, with $z > 0$.

Let's break it down step by step:

1. Simplify each square root:

   • $20\sqrt{z^6}$: The square root of $z^6$ is $z^3$ because $\sqrt{z^6} = z^3$. Therefore, $20\sqrt{z^6} = 20z^3$.
   • $\sqrt{16z^7}$: This can be split into $\sqrt{16}$ and $\sqrt{z^7}$. The square root of 16 is 4, and $\sqrt{z^7} = z^3 \cdot \sqrt{z}$. So, $\sqrt{16z^7} = 4z^3\sqrt{z}$.

2. Set up the quotient:

   $\frac{20z^3}{4z^3\sqrt{z}}$

3. Simplify the fraction:

   • The $z^3$ in the numerator and denominator cancels out.
   • $\frac{20}{4} = 5$, so the quotient simplifies to:

   $\frac{20z^3}{4z^3\sqrt{z}} = \frac{5}{\sqrt{z}}$

4. Rationalize the denominator (if necessary):

   To rationalize the denominator, multiply both the numerator and the denominator by $\sqrt{z}$:

   $\frac{5}{\sqrt{z}} \times \frac{\sqrt{z}}{\sqrt{z}} = \frac{5\sqrt{z}}{z}$

So, the simplified result is $\frac{5\sqrt{z}}{z}$.

Would you like more details on any step, or do you have any questions?

Here are 5 related questions:

1. How do you simplify expressions involving square roots and variables?
2. Why is it important to rationalize the denominator in radical expressions?
3. Can you think of another way to simplify $\frac{20z^3}{4z^3\sqrt{z}}$?
4. What would change if the exponent of $z$ was an odd number in either the numerator or denominator?
5. How would you handle the simplification if the base inside the radical had a negative value?

Tip: Always simplify square roots first before handling division or multiplication in radical expressions.