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Let's simplify the given mathematical expression step by step. The expression is:

\[
\sqrt[4]{16^2} \cdot 4^5 \cdot \left( 4^5 \right)^{\frac{2}{3}} \cdot \frac{4^3 \cdot 4^7}{4^5 \cdot 4^7}
\]

### Step 1: Simplify the Root Term
The first part is \( \sqrt[4]{16^2} \).

- \( 16 = 2^4 \), so \( 16^2 = (2^4)^2 = 2^8 \).
- Now, take the 4th root: \( \sqrt[4]{2^8} = 2^2 = 4 \).

Thus, \( \sqrt[4]{16^2} = 4 \).

### Step 2: Exponent Simplification
Now, simplify \( \left( 4^5 \right)^{\frac{2}{3}} \).

- Use the power of a power rule: \( \left( a^m \right)^n = a^{m \cdot n} \).
- So, \( \left( 4^5 \right)^{\frac{2}{3}} = 4^{5 \cdot \frac{2}{3}} = 4^{\frac{10}{3}} \).

### Step 3: Combine Like Terms
Now, let's focus on combining the terms involving powers of 4.

\[
4^5 \cdot 4^{\frac{10}{3}} \cdot \frac{4^3 \cdot 4^7}{4^5 \cdot 4^7}
\]

Using the laws of exponents, \( a^m \cdot a^n = a^{m+n} \), and for division, \( a^m / a^n = a^{m-n} \), we proceed as follows:

1. Multiply the first two terms: 
   \[
   4^5 \cdot 4^{\frac{10}{3}} = 4^{5 + \frac{10}{3}} = 4^{\frac{15}{3} + \frac{10}{3}} = 4^{\frac{25}{3}}
   \]

2. Now simplify the fraction:
   \[
   \frac{4^3 \cdot 4^7}{4^5 \cdot 4^7} = \frac{4^{3+7}}{4^{5+7}} = \frac{4^{10}}{4^{12}} = 4^{10-12} = 4^{-2}
   \]

### Step 4: Final Combination
Now combine the result from both parts:

\[
4^{\frac{25}{3}} \cdot 4^{-2}
\]

Using the rule \( a^m \cdot a^n = a^{m+n} \), we get:

\[
4^{\frac{25}{3} + (-2)} = 4^{\frac{25}{3} - \frac{6}{3}} = 4^{\frac{19}{3}}
\]

### Final Answer:
The simplified expression is:

\[
4^{\frac{19}{3}}
\]

Let me know if you need further clarifications, or if you'd like to dive deeper into any step!

Here are 5 related questions to expand your understanding:
1. How do you simplify expressions involving fractional exponents?
2. What are the properties of exponents when adding or subtracting powers?
3. How does the rule \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \) apply to this problem?
4. How would this process change if the numbers involved were not powers of 4?
5. Can you generalize the steps for simplifying any expression with multiple exponents?

**Tip**: When simplifying powers, always start by grouping terms with the same base and applying exponent rules to avoid confusion.

Simplifica la siguiente expresión matemática: \sqrt[4]{16^2} \cdot 4^5 \cdot (4^5)^{\frac{2}{3}} \cdot \frac{4^3 \cdot 4^7}{4^5 \cdot 4^7}

Simplify the complex expression involving roots and powers: \sqrt[4]{16^2} \cdot 4^5 \cdot (4^5)^{\frac{2}{3}} \cdot \frac{4^3 \cdot 4^7}{4^5 \cdot 4^7}. Learn the step-by-step solution using exponent laws.

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