1. **State the problem:** Simplify the expression $$\left(\frac{n^{2} \cdot n^{4}}{(2n^{4})^{3} n^{4}}\right)^{4}$$. 2. **Recall exponent rules:** - When multiplying like bases, add exponents: $$a^{m} \cdot a^{n} = a^{m+n}$$. - When dividing like bases, subtract exponents: $$\frac{a^{m}}{a^{n}} = a^{m-n}$$. - Power of a power: $$(a^{m})^{n} = a^{m \cdot n}$$. - Power of a product: $$(ab)^{n} = a^{n} b^{n}$$. 3. **Simplify numerator:** $$n^{2} \cdot n^{4} = n^{2+4} = n^{6}$$. 4. **Simplify denominator inside parentheses:** First, simplify $$(2n^{4})^{3}$$: $$2^{3} \cdot (n^{4})^{3} = 8 n^{12}$$. So denominator is: $$8 n^{12} \cdot n^{4} = 8 n^{12+4} = 8 n^{16}$$. 5. **Rewrite the fraction inside parentheses:** $$\frac{n^{6}}{8 n^{16}} = \frac{1}{8} \cdot \frac{n^{6}}{n^{16}} = \frac{1}{8} n^{6-16} = \frac{1}{8} n^{-10}$$. 6. **Use \cancel to show simplification:** $$\frac{n^{\cancel{6}}}{8 n^{\cancel{6}+10}} = \frac{1}{8} n^{-10}$$. 7. **Raise the entire expression to the 4th power:** $$\left(\frac{1}{8} n^{-10}\right)^{4} = \left(\frac{1}{8}\right)^{4} \cdot (n^{-10})^{4} = \frac{1}{8^{4}} n^{-40}$$. 8. **Calculate $8^{4}$:** $$8^{4} = (2^{3})^{4} = 2^{12} = 4096$$. 9. **Final simplified expression:** $$\frac{1}{4096} n^{-40} = \frac{1}{4096 n^{40}}$$. **Answer:** $$\boxed{\frac{1}{4096 n^{40}}}$$