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.

Learn how to simplify the expression [2^(1/4)/6z]*[z^(5/4)/6z] with a step-by-step solution. This math problem involves key algebraic concepts such as exponent rules and fraction multiplication. Suitable for students in Grades 9-12.

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