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Let's solve the given equation step-by-step.

The given equation is:

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

First, we will simplify both sides of the equation.

### Left Side Simplification

\[
(7^5 \cdot 7^n)^4
\]

Since the bases are the same, we can add the exponents:

\[
(7^{5+n})^4
\]

Now, applying the power rule \((a^m)^n = a^{m \cdot n}\):

\[
7^{(5+n) \cdot 4} = 7^{4(5+n)} = 7^{20 + 4n}
\]

### Right Side Simplification

\[
\left(\frac{7^{16}}{7^n}\right)^3
\]

First, simplify the fraction by subtracting the exponents:

\[
\left(7^{16-n}\right)^3
\]

Now, applying the power rule \((a^m)^n = a^{m \cdot n}\):

\[
7^{(16-n) \cdot 3} = 7^{48 - 3n}
\]

### Equating Both Sides

Now, we have:

\[
7^{20 + 4n} = 7^{48 - 3n}
\]

Since the bases are the same, we can set the exponents equal to each other:

\[
20 + 4n = 48 - 3n
\]

Solve for \(n\):

\[
20 + 4n = 48 - 3n
\]

Add \(3n\) to both sides:

\[
20 + 4n + 3n = 48
\]

Combine like terms:

\[
20 + 7n = 48
\]

Subtract 20 from both sides:

\[
7n = 28
\]

Divide both sides by 7:

\[
n = 4
\]

### Conclusion

The value of \(n\) that satisfies the given equation is \(n = 4\).

Do you want any further details or have any questions?

Here are 5 related questions to further your understanding:

1. How do you handle equations with different bases on each side?
2. What are the steps to simplify expressions with exponents and fractions?
3. Can you explain the power rule in more detail?
4. How can you verify if a solution to an exponent equation is correct?
5. What are some real-world applications of solving equations with exponents?

**Tip:** When dealing with equations involving exponents, always remember to first simplify each side of the equation separately before setting the exponents equal to each other. This helps in avoiding mistakes and makes the process clearer.

Explore the detailed solution to the equation (7^5 · 7^n)^4 = (7^16 / 7^n)^3, focusing on the properties of exponents and algebraic simplification. Suitable for students in Grades 9-12.

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