1. **State the problem:** Simplify the expression $$\frac{16^{\frac{3}{4}n}}{8^{\frac{5}{3}}} \times 4^{n-1}$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$a^m \div a^n = a^{m-n}$$. - When multiplying powers with the same base, add exponents: $$a^m \times a^n = a^{m+n}$$. - Express all bases as powers of a common base if possible. 3. **Rewrite bases as powers of 2:** - $$16 = 2^4$$ - $$8 = 2^3$$ - $$4 = 2^2$$ 4. **Rewrite the expression:** $$\frac{(2^4)^{\frac{3}{4}n}}{(2^3)^{\frac{5}{3}}} \times (2^2)^{n-1}$$ 5. **Simplify exponents:** - $$ (2^4)^{\frac{3}{4}n} = 2^{4 \times \frac{3}{4}n} = 2^{3n} $$ - $$ (2^3)^{\frac{5}{3}} = 2^{3 \times \frac{5}{3}} = 2^5 $$ - $$ (2^2)^{n-1} = 2^{2(n-1)} = 2^{2n - 2} $$ 6. **Substitute back:** $$ \frac{2^{3n}}{2^5} \times 2^{2n - 2} $$ 7. **Divide powers:** $$ 2^{3n - 5} \times 2^{2n - 2} $$ 8. **Multiply powers:** $$ 2^{(3n - 5) + (2n - 2)} = 2^{5n - 7} $$ **Final answer:** $$2^{5n - 7}$$