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CAMBIAR LA N CON EL NUMERO 7 Y RESOLVER PROBLEMA

To solve the problem, let's first analyze and interpret the mathematical expression from the image.

The expression is:

\[
\left(\frac{\frac{1^n \cdot n^n}{5} - \frac{2}{3n}}{\frac{3}{2n} \cdot \frac{1^n}{9}}\right)^3
\]

We are asked to replace \( n \) with the number 7, and then simplify the expression to the maximum.

### Step 1: Substitute \( n = 7 \)

Let's replace \( n \) with 7 in the expression:

\[
\left(\frac{\frac{1^7 \cdot 7^7}{5} - \frac{2}{3 \cdot 7}}{\frac{3}{2 \cdot 7} \cdot \frac{1^7}{9}}\right)^3
\]

### Step 2: Simplify the Expression

1. **Simplify \( 1^7 \):** Since any number raised to the power of 1 is 1, we have:

   \[
   1^7 = 1
   \]

2. **Calculate \( 7^7 \):**

   \[
   7^7 = 823543
   \]

3. **Simplify the numerator:**

   \[
   \frac{1 \cdot 823543}{5} - \frac{2}{21}
   \]

   Simplifying each term:
   
   \[
   \frac{823543}{5} = 164708.6
   \]
   
   The second term is already simplified: \( \frac{2}{21} \).

   So, the numerator becomes:
   
   \[
   164708.6 - \frac{2}{21}
   \]

   To combine these terms, find a common denominator:

   \[
   164708.6 - \frac{2}{21} = \frac{164708.6 \cdot 21}{21} - \frac{2}{21} = \frac{3458880.6 - 2}{21} = \frac{3458878.6}{21}
   \]

4. **Simplify the denominator:**

   \[
   \frac{3}{14} \cdot \frac{1}{9} = \frac{3}{126} = \frac{1}{42}
   \]

5. **Simplify the entire expression:**

   \[
   \frac{\frac{3458878.6}{21}}{\frac{1}{42}} = \frac{3458878.6}{21} \times 42 = 164708.6 \times 42 = 6917757.2
   \]

### Step 3: Raise the Result to the Power of 3

Finally, we raise the result to the power of 3:

\[
6917757.2^3 \approx 3.307 \times 10^{20}
\]

### Final Answer

The fully simplified expression when \( n = 7 \) is approximately:

\[
\left(\frac{3458878.6}{21} \times 42\right)^3 \approx 3.307 \times 10^{20}
\]

---

Would you like more details or have any questions on this problem? Here are some questions that might expand your understanding:

1. What is the general process of simplifying a fraction that involves algebraic expressions?
2. How can you quickly estimate large powers without a calculator?
3. How do you find a common denominator for fractions with large numbers?
4. What are some properties of exponents that are useful in simplifying such expressions?
5. How would the final result change if \( n \) were replaced by a different number?

**Tip:** When simplifying complex fractions, always try to break down the problem into smaller steps, and simplify as you go along to avoid dealing with very large or very small numbers directly.

Learn how to simplify a complex mathematical expression involving exponents and fractions. Follow a step-by-step solution to replace variables and simplify the expression effectively. Suitable for high school students studying algebra.

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