1. **State the problem:** Simplify the expression $$3^{n+1} \times 9^n \div 27^{\frac{2}{3}n}$$. 2. **Recall the base conversions:** - $9 = 3^2$ - $27 = 3^3$ 3. **Rewrite all terms with base 3:** - $3^{n+1}$ stays as is. - $9^n = (3^2)^n = 3^{2n}$. - $27^{\frac{2}{3}n} = (3^3)^{\frac{2}{3}n} = 3^{3 \times \frac{2}{3}n} = 3^{2n}$. 4. **Substitute back into the expression:** $$3^{n+1} \times 3^{2n} \div 3^{2n}$$ 5. **Apply exponent rules:** - Multiplying powers with the same base: $a^m \times a^n = a^{m+n}$ - Dividing powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$ 6. **Simplify numerator:** $$3^{n+1} \times 3^{2n} = 3^{(n+1) + 2n} = 3^{3n + 1}$$ 7. **Divide by denominator:** $$\frac{3^{3n + 1}}{3^{2n}} = 3^{(3n + 1) - 2n} = 3^{n + 1}$$ **Final answer:** $$3^{n+1}$$