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The worksheet you're working on is focused on *rational exponents* and asks for the simplification of expressions without leaving negative exponents in the final answers. Let's break down and solve the two problems shown in the image.

### Problem 1:
\[
\left( (4x^2 y)^{\frac{1}{2}} \left( \frac{x^{\frac{3}{2}}}{2y^{-1}} \right)^{-1} \right)^3
\]

We need to:
1. Simplify each term inside the parentheses.
2. Apply the exponents correctly.

#### Step-by-Step Solution:
1. Simplify \( (4x^2 y)^{\frac{1}{2}} \):
   - The square root exponent \(\frac{1}{2}\) applies to each factor:
     \[
     (4x^2 y)^{\frac{1}{2}} = 2x y^{\frac{1}{2}}
     \]

2. Simplify \( \left( \frac{x^{\frac{3}{2}}}{2y^{-1}} \right)^{-1} \):
   - The negative exponent \(-1\) inverts the fraction:
     \[
     \left( \frac{x^{\frac{3}{2}}}{2y^{-1}} \right)^{-1} = \frac{2y^{-1}}{x^{\frac{3}{2}}}
     \]
   - Apply the negative exponent to \( y \), making it positive:
     \[
     = \frac{2y}{x^{\frac{3}{2}}}
     \]

3. Multiply the results:
   \[
   2x y^{\frac{1}{2}} \cdot \frac{2y}{x^{\frac{3}{2}}} = \frac{4xy^{\frac{3}{2}}}{x^{\frac{3}{2}}}
   \]

4. Simplify the exponents:
   - \( \frac{4x^{1}}{x^{\frac{3}{2}}} = 4x^{-\frac{1}{2}} = \frac{4}{x^{\frac{1}{2}}} \)
   - Combine with the \( y \) terms:
     \[
     \frac{4}{x^{\frac{1}{2}}} \cdot y^{\frac{3}{2}} = \frac{4y^{\frac{3}{2}}}{x^{\frac{1}{2}}}
     \]

5. Now apply the cube exponent \( 3 \) to the entire expression:
   \[
   \left( \frac{4y^{\frac{3}{2}}}{x^{\frac{1}{2}}} \right)^3 = \frac{4^3 y^{\frac{9}{2}}}{x^{\frac{3}{2}}}
   \]
   Simplify:
   \[
   \frac{64y^{\frac{9}{2}}}{x^{\frac{3}{2}}}
   \]

Final Answer for Problem 1:
\[
\frac{64y^{\frac{9}{2}}}{x^{\frac{3}{2}}}
\]

---

### Problem 2:
\[
\left( \frac{a^{\frac{1}{2}} b^{\frac{3}{2}}}{a^{-1}b^{-1}c} \right)^{-2} \cdot \frac{c^3}{a^3}
\]

#### Step-by-Step Solution:
1. Simplify the first fraction inside the parentheses:
   \[
   \frac{a^{\frac{1}{2}} b^{\frac{3}{2}}}{a^{-1}b^{-1}c} = a^{\frac{1}{2} - (-1)} b^{\frac{3}{2} - (-1)} c^{-1}
   \]
   Simplify the exponents:
   \[
   = a^{\frac{3}{2}} b^{\frac{5}{2}} c^{-1}
   \]

2. Apply the exponent \(-2\):
   \[
   \left( a^{\frac{3}{2}} b^{\frac{5}{2}} c^{-1} \right)^{-2} = a^{-3} b^{-5} c^{2}
   \]

3. Multiply by \( \frac{c^3}{a^3} \):
   \[
   a^{-3} b^{-5} c^2 \cdot \frac{c^3}{a^3} = a^{-3-3} b^{-5} c^{2+3}
   \]
   Simplify:
   \[
   = a^{-6} b^{-5} c^5
   \]

4. Rewrite without negative exponents:
   \[
   \frac{c^5}{a^6 b^5}
   \]

Final Answer for Problem 2:
\[
\frac{c^5}{a^6 b^5}
\]

---

Let me know if you'd like further clarification or more details!

### Related Questions:
1. What are the steps for simplifying expressions with fractional exponents?
2. How can negative exponents be removed in more complex expressions?
3. What is the significance of simplifying radical expressions with variables?
4. How do exponent rules apply when multiple variables are involved in a fraction?
5. Can these exponent simplifications be applied to real-world problems?

### Tip:
Always ensure that when you simplify expressions with negative exponents, you rewrite them as positive exponents by inverting the base, making it easier to work with final simplified forms.

Simplify expressions involving rational exponents. No negative exponents should remain in the final result.

Learn how to simplify expressions involving rational exponents, ensuring no negative exponents are left in the final result. This tutorial covers exponent rules, including multiplication, division, and negative exponent simplification, suitable for high school students.

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