You're on a roll — i) Step 1: Simplify $8^{1/3}$. $8^{1/3} = \sqrt[3]{8} = 2$. Step 2: Rewrite the expression using the simplified term and express all numbers as powers of 2. $8^{1/3} \times 8^{-3} \times 32 = 2 \times (2^3)^{-3} \times 2^5$ Step 3: Apply the exponent rule $(a^m)^n = a^{mn}$. $2 \times 2^{3 \times (-3)} \times 2^5 = 2^1 \times 2^{-9} \times 2^5$ Step 4: Apply the exponent rule $a^m \times a^n = a^{m+n}$. $2^{1 + (-9) + 5} = 2^{1 - 9 + 5} = 2^{-3}$ Step 5: Calculate the final value. $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ The simplified expression is $\boxed{\frac{1}{8}}$. ii) Step 1: Express all numbers in the expression as powers of 3. $27 = 3^3$ $9 = 3^2$ $81 = 3^4$ Substitute these into the expression: $$ \frac{(3^3)^{2/3} \times (3^2)^5}{(3^4)^4} $$ Step 2: Apply the exponent rule $(a^m)^n = a^{mn}$ to each term. $$ \frac{3^{3 \times (2/3)} \times 3^{2 \times 5}}{3^{4 \times 4}} = \frac{3^2 \times 3^{10}}{3^{16}} $$ Step 3: Apply the exponent rule $a^m \times a^n = a^{m+n}$ to the numerator. $$ \frac{3^{2+10}}{3^{16}} = \frac{3^{12}}{3^{16}} $$ Step 4: Apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$. $$ 3^{12-16} = 3^{-4} $$ Step 5: Calculate the final value. $$ 3^{-4} = \frac{1}{3^4} = \frac{1}{81} $$ The simplified expression is $\boxed{\frac{1}{81}}$. Got more? Send 'em.