1. **State the problem:** We are given the equation $$1 \times 2^{3/1} = 8^x$$ and the expression $$4^n$$, and we want to express $$x$$ in terms of $$n$$. 2. **Rewrite the bases as powers of 2:** Note that $$8 = 2^3$$ and $$4 = 2^2$$. 3. **Rewrite the equation using powers of 2:** $$1 \times 2^{3} = (2^3)^x$$ which simplifies to $$2^3 = 2^{3x}$$. 4. **Equate the exponents:** Since the bases are the same, the exponents must be equal: $$3 = 3x \implies x = 1$$. 5. **Express $$4^n$$ in terms of powers of 2:** $$4^n = (2^2)^n = 2^{2n}$$. 6. **Relate the expressions involving $$x$$ and $$n$$:** From the user's notes, it seems they want to relate expressions like $$2^{1/2} \times 2^{3/1}$$ and $$2^{2}n - 2n$$, but the key is to express $$x$$ in terms of $$n$$ from the original equation. 7. **Summary:** The original equation simplifies to $$x = 1$$, which is independent of $$n$$. **Final answer:** $$x = 1$$