1. **State the problem:** Calculate the value of $$81^{\frac{3}{4}} \times 9^{\frac{1}{2}}$$. 2. **Recall the rules:** - For any positive number $a$ and rational exponent $\frac{m}{n}$, $$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$. - Simplify powers by expressing the bases as powers of the same number if possible. 3. **Express bases as powers of 3:** - $81 = 3^4$ - $9 = 3^2$ 4. **Rewrite the expression:** $$81^{\frac{3}{4}} \times 9^{\frac{1}{2}} = (3^4)^{\frac{3}{4}} \times (3^2)^{\frac{1}{2}}$$ 5. **Use power of a power rule:** $$ (a^m)^n = a^{m \times n} $$ 6. **Calculate each term:** $$ (3^4)^{\frac{3}{4}} = 3^{4 \times \frac{3}{4}} = 3^3 $$ $$ (3^2)^{\frac{1}{2}} = 3^{2 \times \frac{1}{2}} = 3^1 = 3 $$ 7. **Multiply the results:** $$ 3^3 \times 3 = 3^{3+1} = 3^4 $$ 8. **Calculate the final value:** $$ 3^4 = 81 $$ **Final answer:** $$81^{\frac{3}{4}} \times 9^{\frac{1}{2}} = 81$$