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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.

Use the properties of exponents to simplify the expression: (y^(3/2) * x^(-1/2))^4.

Learn how to simplify the expression (y^(3/2) * x^(-1/2))^4 by applying the power of a power property and other exponent rules. This algebraic problem covers essential exponentiation concepts, suitable for high school students (Grades 9-12).

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