[2(4x^-5/3 . x^2 + y) / (8y^-4/7 (3x)^-10/3) ]^2

We are tasked with simplifying the following expression:

\[
\left[\frac{2\left(\frac{4x^{-\frac{5}{3}} \cdot x^2 + y}{8y^{-\frac{4}{7}} \cdot (3x)^{-\frac{10}{3}}}\right)}\right]^2
\]

### Step-by-Step Simplification

1. **Simplify the terms inside the parentheses**:
   - Start with the numerator:
     \[
     4x^{-\frac{5}{3}} \cdot x^2 = 4x^{-\frac{5}{3} + 2}
     \]
     To combine the exponents, express 2 as \(\frac{6}{3}\):
     \[
     4x^{-\frac{5}{3} + \frac{6}{3}} = 4x^{\frac{1}{3}}
     \]
     So, the numerator becomes:
     \[
     4x^{\frac{1}{3}} + y
     \]

2. **Simplify the denominator**:
   \[
   8y^{-\frac{4}{7}} \cdot (3x)^{-\frac{10}{3}}
   \]
   First, simplify \((3x)^{-\frac{10}{3}}\):
   \[
   (3x)^{-\frac{10}{3}} = 3^{-\frac{10}{3}} \cdot x^{-\frac{10}{3}}
   \]
   Thus, the denominator becomes:
   \[
   8y^{-\frac{4}{7}} \cdot 3^{-\frac{10}{3}} \cdot x^{-\frac{10}{3}}
   \]

3. **Write the expression inside the brackets**:
   The fraction simplifies to:
   \[
   \frac{4x^{\frac{1}{3}} + y}{8y^{-\frac{4}{7}} \cdot 3^{-\frac{10}{3}} \cdot x^{-\frac{10}{3}}}
   \]

4. **Factor out constants and powers of 2 and 3**:
   The constant factors can be simplified:
   \[
   \frac{2 \cdot (4x^{\frac{1}{3}} + y)}{8 \cdot 3^{-\frac{10}{3}} \cdot y^{-\frac{4}{7}} \cdot x^{-\frac{10}{3}}}
   \]
   Simplify the constant term \( \frac{2}{8} = \frac{1}{4} \), leaving:
   \[
   \frac{4x^{\frac{1}{3}} + y}{4 \cdot 3^{-\frac{10}{3}} \cdot y^{-\frac{4}{7}} \cdot x^{-\frac{10}{3}}}
   \]

   This simplifies further to:
   \[
   \frac{x^{\frac{1}{3}} + y}{3^{-\frac{10}{3}} \cdot y^{-\frac{4}{7}} \cdot x^{-\frac{10}{3}}}
   \]

5. **Apply the exponent outside the parentheses**:
   We now square the entire expression:
   \[
   \left( \frac{x^{\frac{1}{3}} + y}{3^{-\frac{10}{3}} \cdot y^{-\frac{4}{7}} \cdot x^{-\frac{10}{3}}} \right)^2
   \]

   This leads to:
   \[
   \frac{(x^{\frac{1}{3}} + y)^2}{(3^{-\frac{10}{3}} \cdot y^{-\frac{4}{7}} \cdot x^{-\frac{10}{3}})^2}
   \]
   Now, square the denominator:
   \[
   3^{-\frac{20}{3}} \cdot y^{-\frac{8}{7}} \cdot x^{-\frac{20}{3}}
   \]

6. **Final simplified expression**:
   The final result is:
   \[
   \frac{(x^{\frac{1}{3}} + y)^2}{3^{-\frac{20}{3}} \cdot y^{-\frac{8}{7}} \cdot x^{-\frac{20}{3}}}
   \]

Let me know if you need further details or have any questions.

### Related Questions:
1. How do you combine exponents with the same base in algebra?
2. What is the process for distributing exponents across a fraction?
3. How do you simplify expressions with negative exponents?
4. What are the common laws of exponents used in simplification?
5. How do you handle fractions raised to a power?

### Tip:
When dealing with negative exponents, remember that \( a^{-n} = \frac{1}{a^n} \), and this can help simplify expressions quickly.

Explore the step-by-step simplification of the algebraic expression [2(4x^-5/3 . x^2 + y) / (8y^-4/7 (3x)^-10/3)]^2. This detailed solution covers exponent rules, algebraic simplification, and handling fractions with negative exponents, suitable for high school students in Grades 10-12.

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