Understanding the Equation
To solve the equation \([ \sqrt{2} \cdot (16^{1/4} + 7776^{1/5})^4 ]^n = 4\sqrt{2}\), we need to simplify both sides.

Step 1: Simplifying the Left Side
- Calculate \(16^{1/4}\):
- \(16 = 2^4\)
- Thus, \(16^{1/4} = (2^4)^{1/4} = 2^{4/4} = 2\)
- Calculate \(7776^{1/5}\):
- Factor \(7776\) to find \(7776 = 6^5\)
- Therefore, \(7776^{1/5} = (6^5)^{1/5} = 6\)
- Combine these results:
- \(16^{1/4} + 7776^{1/5} = 2 + 6 = 8\)
- Now, plug this back into the equation:
- \((16^{1/4} + 7776^{1/5})^4 = 8^4 = 4096\)
- Thus, the left side becomes:
- \(\sqrt{2} \cdot 4096 = 4096\sqrt{2}\)

Step 2: Simplifying the Right Side
- The right side is \(4\sqrt{2}\).

Step 3: Setting the Equation
Now we have:
\[
(4096\sqrt{2})^n = 4\sqrt{2}
\]

Step 4: Equating and Solving
- Rewrite \(4096\) as \(2^{12}\):
- Therefore, \((2^{12}\sqrt{2})^n = 2^{12n + 1/2}\)
- Now, express \(4\sqrt{2}\):
- \(4\sqrt{2} = 2^2 \cdot 2^{1/2} = 2^{2.5} = 2^{5/2}\)
Set the exponents equal:
\[
12n + \frac{1}{2} = \frac{5}{2}
\]
- Solve for \(n\):
- \(12n = \frac{5}{2} - \frac{1}{2} = \frac{4}{2} = 2\)
- \(n = \frac{2}{12} = \frac{1}{6}\)

Final Answer
The value of \(n\) is \(\frac{1}{6}\).