\( \frac { 3 ^ { 3 i } \times 27 ^ { 3 y } \times 3 ^ { 6 } } { 81 ^ { 3 y } } \)

We start with the expression: \[ \frac{3^{3i} \times 27^{3y} \times 3^6}{81^{3y}} \] **Step 1. Express all bases as powers of 3.** Recall that: \[ 27 = 3^3 \quad \text{and} \quad 81 = 3^4 \] **Step 2. Rewrite the expression using these equivalences.** Substitute into the expression: \[ \frac{3^{3i} \times (3^3)^{3y} \times 3^6}{(3^4)^{3y}} \] **Step 3. Simplify the exponents in the numerator and the denominator.** Apply the power rule \((a^b)^c = a^{bc}\): \[ (3^3)^{3y} = 3^{9y} \quad \text{and} \quad (3^4)^{3y} = 3^{12y} \] The expression becomes: \[ \frac{3^{3i} \times 3^{9y} \times 3^6}{3^{12y}} \] **Step 4. Combine the exponents in the numerator.** Using the product rule \(a^m \times a^n = a^{m+n}\): \[ 3^{3i} \times 3^{9y} \times 3^6 = 3^{3i + 9y + 6} \] The fraction is now: \[ \frac{3^{3i + 9y + 6}}{3^{12y}} \] **Step 5. Simplify the fraction by subtracting the exponents.** Using the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\): \[ 3^{3i + 9y + 6 - 12y} = 3^{3i - 3y + 6} \] The final simplified expression is: \[ 3^{3i - 3y + 6} \]