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.