change the -3/1 and -2/1 into -1/3 and -1/2

Got it! Let's update the expression as you requested. Here's the reformulated expression:

\[
\left\{ x^{\frac{2}{3}} \left[ x^{-\frac{1}{3}} y^{-\frac{1}{2}} (x^2 y^{-2})^{-\frac{3}{2}} \right]^{\frac{1}{2}} \right\}^6
\]

I'll now proceed with simplifying this step by step.

### Step-by-step simplification:

1. **Simplify the inner expression** \((x^2 y^{-2})^{-\frac{3}{2}}\):

\[
(x^2 y^{-2})^{-\frac{3}{2}} = x^{2 \cdot -\frac{3}{2}} y^{-2 \cdot -\frac{3}{2}} = x^{-3} y^3
\]

Now the expression becomes:

\[
\left\{ x^{\frac{2}{3}} \left[ x^{-\frac{1}{3}} y^{-\frac{1}{2}} x^{-3} y^3 \right]^{\frac{1}{2}} \right\}^6
\]

2. **Combine the powers of \(x\) and \(y\) within the square brackets**:

\[
x^{-\frac{1}{3}} x^{-3} = x^{-\frac{1}{3} - 3} = x^{-\frac{10}{3}}
\]
\[
y^{-\frac{1}{2}} y^3 = y^{-\frac{1}{2} + 3} = y^{\frac{5}{2}}
\]

Thus, the expression becomes:

\[
\left\{ x^{\frac{2}{3}} \left[ x^{-\frac{10}{3}} y^{\frac{5}{2}} \right]^{\frac{1}{2}} \right\}^6
\]

3. **Apply the \(\frac{1}{2}\) power to both \(x\) and \(y\)**:

\[
\left[ x^{-\frac{10}{3}} y^{\frac{5}{2}} \right]^{\frac{1}{2}} = x^{-\frac{5}{3}} y^{\frac{5}{4}}
\]

Now the expression is:

\[
\left\{ x^{\frac{2}{3}} x^{-\frac{5}{3}} y^{\frac{5}{4}} \right\}^6
\]

4. **Combine the powers of \(x\)**:

\[
x^{\frac{2}{3}} x^{-\frac{5}{3}} = x^{\frac{2}{3} - \frac{5}{3}} = x^{-\frac{3}{3}} = x^{-1}
\]

Thus, the expression simplifies to:

\[
\left\{ x^{-1} y^{\frac{5}{4}} \right\}^6
\]

5. **Apply the power of 6** to both \(x\) and \(y\)**:

\[
x^{-1 \cdot 6} y^{\frac{5}{4} \cdot 6} = x^{-6} y^{\frac{30}{4}} = x^{-6} y^{\frac{15}{2}}
\]

### Final simplified expression:

\[
x^{-6} y^{\frac{15}{2}}
\]

This is the simplified version of the original expression.

---

Do you have any questions or would you like more details about any of the steps?

Here are 5 related questions for further exploration:
1. What happens when you change the exponents in a similar expression?
2. How would you simplify expressions involving negative exponents differently?
3. Can you explain the rules for combining powers when multiplying exponents?
4. What is the importance of fractional exponents in algebra?
5. How does simplifying an expression with exponents affect its graph?

**Tip**: Always apply exponent rules carefully, especially when dealing with fractions and negative values.

Understand how to change exponents from -3/1 and -2/1 to -1/3 and -1/2 in mathematical expressions. Detailed step-by-step explanation with examples and simplification. Suitable for high school students and beyond.

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