The expression given is $\left( y^{\frac{3}{2}} x^{-\frac{1}{2}} \right)^4$.

To simplify this expression, use the power of a power property, which states that $(a^m)^n = a^{m \cdot n}$.

Step-by-step process:

1. Apply the exponent to each term inside the parentheses: $\left( y^{\frac{3}{2}} \right)^4 = y^{\frac{3}{2} \cdot 4} = y^6$ $\left( x^{-\frac{1}{2}} \right)^4 = x^{-\frac{1}{2} \cdot 4} = x^{-2}$

2. Combine the results: $y^6 x^{-2}$

Thus, the simplified expression is: $y^6 x^{-2}$

Would you like further details or have any other questions?

Related questions:

1. How does the property $(a^m)^n = a^{m \cdot n}$ work in other cases?
2. What happens if there are more variables inside the parentheses?
3. How would the result change if the exponents were fractions other than $\frac{3}{2}$ and $-\frac{1}{2}$?
4. How can you rewrite expressions with negative exponents as positive exponents?
5. How does this simplification relate to multiplying and dividing powers of the same base?

Tip: When simplifying expressions involving exponents, always look for opportunities to use exponent rules, such as the power of a power and negative exponent rules.