To simplify the expression $\left(\frac{2^{1/4}}{6z}\right) \times \left(\frac{z^{5/4}}{6z}\right)$, let's follow these steps:

Step 1: Multiply the numerators

$2^{1/4} \times z^{5/4} = 2^{1/4} z^{5/4}$

Step 2: Multiply the denominators

$6z \times 6z = 36z^2$

Step 3: Combine the results

The expression becomes: $\frac{2^{1/4} z^{5/4}}{36z^2}$

Step 4: Simplify the powers of $z$

Now, simplify $z^{5/4}$ over $z^2$. Using the exponent rule $\frac{z^a}{z^b} = z^{a-b}$, we get: $z^{5/4} / z^2 = z^{5/4 - 8/4} = z^{-3/4}$

Step 5: Final simplified form

The final simplified expression is: $\frac{2^{1/4}}{36z^{3/4}}$

This is the simplified result.

Would you like further details on any of the steps, or do you have any questions?

Here are 5 related questions for practice:

1. Simplify $\left(\frac{3^{1/3}}{2x}\right) \times \left(\frac{x^{4/3}}{4x}\right)$.
2. How do you simplify $\frac{z^{7/4}}{z^3}$?
3. Multiply and simplify $\left(\frac{5^{1/5}}{3y}\right) \times \left(\frac{y^{6/5}}{9y}\right)$.
4. Simplify the expression $\frac{x^{2/3}}{x^{5/6}}$.
5. Simplify $\left(\frac{7^{1/2}}{4x}\right) \times \left(\frac{x^{3/2}}{5x}\right)$.

Tip: When simplifying powers, always apply the exponent subtraction rule when dividing terms with the same base.