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The expression provided is:

\[
\left( 2p^{\frac{1}{2}} q^6 \right)^4
\]

To simplify this, we apply the power of a product rule and power of powers rule, which states that for any terms inside parentheses being raised to a power, we distribute the outer exponent to each factor. 

### Step-by-step solution:

1. **Apply the outer exponent (4) to each factor inside the parentheses**:
   \[
   \left( 2p^{\frac{1}{2}} q^6 \right)^4 = 2^4 \cdot \left( p^{\frac{1}{2}} \right)^4 \cdot \left( q^6 \right)^4
   \]

2. **Simplify each part**:
   - \( 2^4 = 16 \)
   - \( \left( p^{\frac{1}{2}} \right)^4 = p^{\frac{1}{2} \cdot 4} = p^2 \)
   - \( \left( q^6 \right)^4 = q^{6 \cdot 4} = q^{24} \)

3. **Put it all together**:
   \[
   16p^2q^{24}
   \]

So, the simplified expression is:

\[
16p^2q^{24}
\]

Would you like further clarification on any part of this? 

Here are some related questions you might find helpful:

1. How does the power of a power rule work in exponents?
2. What is the general rule for simplifying expressions with multiple variables and powers?
3. How can we simplify more complex exponential expressions?
4. How does the distributive property of exponents apply in algebraic simplification?
5. What are the different laws of exponents and their applications?

**Tip:** Always distribute the outer exponent to all terms inside the parentheses when simplifying.

Learn how to simplify the expression \left(2p^{1/2}q^6\right)^4 step by step. This problem demonstrates the power of a product rule and power of a power rule, applying these to simplify the exponents of multiple variables. Suitable for Grades 9-10.

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