Understanding the Equation
To solve the equation \([ \sqrt{2} \cdot (16^{1/4} + 7776^{1/5})^4 ]^n = 4 \sqrt{2}\), we start by simplifying the left-hand side.

Step 1: Simplifying \(16^{1/4}\)
- \(16 = 2^4\)
- Therefore, \(16^{1/4} = (2^4)^{1/4} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2\)

Step 2: Simplifying \(7776^{1/5}\)
- Factor \(7776\):
- \(7776 = 2^5 \cdot 3^5\)
- Thus, \(7776^{1/5} = (2^5 \cdot 3^5)^{1/5} = 2^{5/5} \cdot 3^{5/5} = 2 \cdot 3 = 6\)

Step 3: Combine the results
- Now, we have:
- \(16^{1/4} + 7776^{1/5} = 2 + 6 = 8\)

Step 4: Substitute back into the equation
- The equation now reads:
- \([ \sqrt{2} \cdot 8^4 ]^n = 4 \sqrt{2}\)

Step 5: Calculate \(8^4\)
- \(8 = 2^3\)
- Therefore, \(8^4 = (2^3)^4 = 2^{12}\)

Step 6: Substitute this back
- We have:
- \([ \sqrt{2} \cdot 2^{12} ]^n = 4 \sqrt{2}\)

Step 7: Simplify the left side
- \(\sqrt{2} \cdot 2^{12} = 2^{1/2} \cdot 2^{12} = 2^{12.5} = 2^{25/2}\)
- Thus, \([2^{25/2}]^n = 2^{25n/2}\)

Step 8: Simplify the right side
- \(4 \sqrt{2} = 2^2 \cdot 2^{1/2} = 2^{2.5} = 2^{5/2}\)

Step 9: Set the exponents equal
- \( \frac{25n}{2} = \frac{5}{2}\)
- Solving for \(n\):
- \(25n = 5 \Rightarrow n = \frac{5}{25} = \frac{1}{5}\)

Final Answer
- The value of \(n\) is \(\frac{1}{5}\).