Solve for `y` if `(((1)/(9))^(2y-1) (0.0081)^(1//2) )/( sqrt(243)) = ((1)/(3))^(2y -5) root(3) ((27^(y-1) )/(1000000) ).`

Answer

To solve the equation

\[ \frac{\left(\frac{1}{9}\right)^{2y-1} (0.0081)^{\frac{1}{2}}}{\sqrt{243}} = \left(\frac{1}{3}\right)^{2y - 5} \sqrt{3} \cdot \frac{27^{y-1}}{1000000}, \]

we will convert all numbers to powers of 3.

### Step 1: Rewrite the terms in powers of 3

1. **Convert \( \frac{1}{9} \)**:
\[ \frac{1}{9} = 3^{-2} \]
Therefore,
\[ \left(\frac{1}{9}\right)^{2y-1} = (3^{-2})^{2y-1} = 3^{-2(2y-1)} = 3^{-4y + 2} \]

2. **Convert \( 0.0081 \)**:
\[ 0.0081 = \frac{81}{10000} = \frac{3^4}{10^4} \implies (0.0081)^{\frac{1}{2}} = \left(\frac{3^4}{10^4}\right)^{\frac{1}{2}} = \frac{3^2}{10^2} = \frac{9}{100} \]

3. **Convert \( \sqrt{243} \)**:
\[ 243 = 3^5 \implies \sqrt{243} = 3^{\frac{5}{2}} \]

### Step 2: Substitute back into the equation

Now substituting these values back into the equation:
\[ \frac{3^{-4y + 2} \cdot \frac{9}{100}}{3^{\frac{5}{2}}} = \left(\frac{1}{3}\right)^{2y - 5} \cdot \sqrt{3} \cdot \frac{27^{y-1}}{1000000} \]

### Step 3: Simplify the left side

The left side becomes:
\[ \frac{3^{-4y + 2} \cdot 9}{100 \cdot 3^{\frac{5}{2}}} = \frac{3^{-4y + 2} \cdot 3^2}{100 \cdot 3^{\frac{5}{2}}} = \frac{3^{-4y + 4 - \frac{5}{2}}}{100} = \frac{3^{-4y + \frac{3}{2}}}{100} \]

### Step 4: Simplify the right side

Now simplifying the right side:
1. **Convert \( \left(\frac{1}{3}\right)^{2y - 5} \)**:
\[ \left(\frac{1}{3}\right)^{2y - 5} = 3^{-(2y - 5)} = 3^{5 - 2y} \]

2. **Convert \( 27^{y-1} \)**:
\[ 27 = 3^3 \implies 27^{y-1} = (3^3)^{y-1} = 3^{3(y-1)} = 3^{3y - 3} \]

3. **Combine**:
\[ \sqrt{3} = 3^{\frac{1}{2}} \implies \sqrt{3} \cdot \frac{27^{y-1}}{1000000} = 3^{\frac{1}{2}} \cdot 3^{3y - 3} \cdot \frac{1}{1000000} \]

4. **Convert \( 1000000 \)**:
\[ 1000000 = 10^6 = (2 \cdot 5)^6 = 2^6 \cdot 5^6 \]

### Step 5: Set the equation

Now we have:
\[ \frac{3^{-4y + \frac{3}{2}}}{100} = 3^{5 - 2y} \cdot 3^{3y - 3} \cdot \frac{1}{10^6} \]

### Step 6: Combine the right side

Combining the right side:
\[ 3^{5 - 2y + 3y - 3} \cdot \frac{1}{10^6} = 3^{y + 2} \cdot \frac{1}{10^6} \]

### Step 7: Equate powers of 3

Now equate the powers of 3:
\[ -4y + \frac{3}{2} = y + 2 \]

### Step 8: Solve for \( y \)

Rearranging gives:
\[ -4y - y = 2 - \frac{3}{2} \implies -5y = \frac{1}{2} \implies y = -\frac{1}{10} \]

### Final Answer:
Thus, the solution for \( y \) is:
\[ y = -\frac{1}{10} \]