If ` 3 log sqrtm + 2 log ""^(3) sqrtn = 1. ` find the value of `m^(9) n^(4)` .

Answer

To solve the equation \( 3 \log \sqrt{m} + 2 \log n^{\frac{1}{3}} = 1 \) and find the value of \( m^9 n^4 \), we can follow these steps:

### Step 1: Rewrite the logarithmic expressions
Using the property of logarithms that states \( \log a^b = b \log a \), we can rewrite the logarithmic terms:
\[ \log \sqrt{m} = \log m^{\frac{1}{2}} = \frac{1}{2} \log m \]
\[ \log n^{\frac{1}{3}} = \frac{1}{3} \log n \]
Substituting these into the equation gives:
\[ 3 \left(\frac{1}{2} \log m\right) + 2 \left(\frac{1}{3} \log n\right) = 1 \]

### Step 2: Simplify the equation
Now simplify the equation:
\[ \frac{3}{2} \log m + \frac{2}{3} \log n = 1 \]

### Step 3: Eliminate fractions
To eliminate the fractions, find a common denominator, which is 6:
\[ \frac{3 \cdot 3}{2 \cdot 3} \log m + \frac{2 \cdot 2}{3 \cdot 2} \log n = 1 \]
This simplifies to:
\[ \frac{9}{6} \log m + \frac{4}{6} \log n = 1 \]
Multiplying through by 6 gives:
\[ 9 \log m + 4 \log n = 6 \]

### Step 4: Combine the logarithms
Using the property \( a \log b + c \log d = \log(b^a \cdot d^c) \), we can combine the logarithms:
\[ \log(m^9) + \log(n^4) = 6 \]
This can be rewritten as:
\[ \log(m^9 n^4) = 6 \]

### Step 5: Exponentiate to solve for \( m^9 n^4 \)
To eliminate the logarithm, we exponentiate both sides:
\[ m^9 n^4 = 10^6 \]

### Final Answer
Thus, the value of \( m^9 n^4 \) is:
\[ \boxed{1000000} \]