Let's simplify the expression \([m \times (m)^{-2} \times (n)^{5/3}]^2\).

### Step-by-Step Solution:

1. **Inside the brackets**:
   - First, simplify the powers of \(m\):
     \[
     m \times (m)^{-2} = m^{1 - 2} = m^{-1}.
     \]
   - Now the expression becomes:
     \[
     [m^{-1} \times (n)^{5/3}].
     \]
   
2. **Applying the power of 2 to the entire expression**:
   Now raise everything inside the brackets to the power of 2:
   \[
   (m^{-1} \times (n)^{5/3})^2.
   \]
   - Using the power rule \((a \times b)^n = a^n \times b^n\), distribute the power of 2:
     \[
     = (m^{-1})^2 \times ((n)^{5/3})^2.
     \]
   
3. **Simplify the powers**:
   - \( (m^{-1})^2 = m^{-2} \).
   - \( ((n)^{5/3})^2 = n^{10/3} \).
   
   So, the expression simplifies to:
   \[
   m^{-2} \times n^{10/3}.
   \]

### Final Answer:
\[
m^{-2} \times n^{10/3}.
\]

This is the simplified form of the given expression.

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Do you want further details or have any questions?

Here are 5 questions for deeper exploration:
1. How do you simplify expressions with negative exponents?
2. What is the power rule in exponents, and how does it apply to this expression?
3. How do you handle fractional exponents when simplifying expressions?
4. Can you convert fractional exponents into radicals?
5. What happens when you multiply powers with the same base but different exponents?

**Tip:** Always handle negative exponents by moving terms to the denominator when appropriate!

Learn how to simplify the expression [m x (m)^-2 x (n)^5/3]^2. This step-by-step solution covers key algebraic concepts like the power rule, exponentiation, and simplifying negative exponents. Suitable for students in Grades 9-11.

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