The expression you provided, \(2 \times \frac{4}{3^2} \left(8m^2 \times \frac{4}{3} - \frac{4}{3}\right)\), seems to have some formatting issues that might lead to confusion. Let's clarify the expression step-by-step and simplify it.

### Step 1: Understand the Expression

The expression appears to be:

\[
2 \times \frac{4}{3^2} \left(8m^2 \times \frac{4}{3} - \frac{4}{3}\right)
\]

Breaking it down:

1. \(2 \times \frac{4}{3^2}\)
2. Inside the parentheses: \(8m^2 \times \frac{4}{3} - \frac{4}{3}\)

### Step 2: Simplify the Components

1. **Simplify \( \frac{4}{3^2} \):**
   \[
   \frac{4}{3^2} = \frac{4}{9}
   \]

2. **Calculate \(2 \times \frac{4}{9}\):**
   \[
   2 \times \frac{4}{9} = \frac{8}{9}
   \]

3. **Simplify the expression inside the parentheses:**
   - Multiply \(8m^2 \times \frac{4}{3}\):
   \[
   8m^2 \times \frac{4}{3} = \frac{32m^2}{3}
   \]
   - Subtract \(\frac{4}{3}\):
   \[
   \frac{32m^2}{3} - \frac{4}{3} = \frac{32m^2 - 4}{3} = \frac{4(8m^2 - 1)}{3}
   \]

### Step 3: Combine and Simplify the Entire Expression

Now, combine the results from Step 2:

\[
\frac{8}{9} \times \frac{4(8m^2 - 1)}{3}
\]

This can be simplified as:

\[
\frac{8 \times 4(8m^2 - 1)}{9 \times 3} = \frac{32(8m^2 - 1)}{27}
\]

### Final Answer

The simplified form of the expression is:

\[
\frac{32(8m^2 - 1)}{27}
\]

Would you like a more detailed breakdown or have any questions?

### Related Questions:
1. How to simplify expressions involving fractions and exponents?
2. What are common mistakes to avoid in algebraic simplification?
3. How to distribute multiplication over subtraction in algebraic expressions?
4. What are some techniques to factorize quadratic expressions?
5. How to handle complex fractions in algebra?

### Tip:
Always double-check the order of operations (PEMDAS/BODMAS) when simplifying complex expressions to avoid mistakes!

Learn how to simplify complex algebraic expressions involving fractions and exponents step-by-step. This guide demonstrates the use of the order of operations, fraction multiplication, and exponentiation rules, perfect for students in Grades 8-10.

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