Step 1: Understanding the Concept:
This problem involves indices (exponents). To find the values of \( x \) and \( y \), we should express all terms as powers of the same base, which is 3 in this case.
: Key Formula or Approach:
1. \( a^{mn} = (a^m)^n \).
2. \( \frac{1}{a^n} = a^{-n} \).
3. If \( a^x = a^y \), then \( x = y \) (for \( a>0, a \neq 1 \)).
Step 2: Detailed Explanation:
First, express the middle term as a power of 3:
\[ \frac{1}{27\sqrt{27}} = \frac{1}{3^3 \cdot (3^3)^{1/2}} = \frac{1}{3^3 \cdot 3^{3/2}} = \frac{1}{3^{3 + 1.5}} = \frac{1}{3^{9/2}} = 3^{-9/2} \]
Now, equate this to the first term:
\[ 9^{-x} = (3^2)^{-x} = 3^{-2x} \]
\[ 3^{-2x} = 3^{-9/2} \implies -2x = -\frac{9}{2} \implies x = \frac{9}{4} \]
Next, equate it to the third term:
\[ (81)^{-y} = (3^4)^{-y} = 3^{-4y} \]
\[ 3^{-4y} = 3^{-9/2} \implies -4y = -\frac{9}{2} \implies y = \frac{9}{8} \]
Therefore, the pair \( (x, y) = (9/4, 9/8) \).
Step 3: Final Answer:
The value of \( (x, y) \) is \( (9/4, 9/8) \).