A radioactive sample decays with a constant of `(1)/(3)log_(e)2s^(-1)`. If initially there are 200 nuclei present, find the number of nuclei decayed during the first 9 seconds.

Explanation

To solve the problem, we will use the radioactive decay formula:

\[ n = n_0 e^{-\lambda t} \]

where:
- \( n \) is the number of nuclei remaining after time \( t \),
- \( n_0 \) is the initial number of nuclei,
- \( \lambda \) is the decay constant,
- \( t \) is the time.

### Step-by-Step Solution:

1. **Identify the given values**:
- Initial number of nuclei, \( n_0 = 200 \)
- Decay constant, \( \lambda = \frac{1}{3} \ln 2 \, \text{s}^{-1} \)
- Time, \( t = 9 \, \text{s} \)

2. **Substitute the values into the decay formula**:
\[ n = 200 e^{-\left(\frac{1}{3} \ln 2\right) \cdot 9} \]

3. **Calculate the exponent**:
\[ -\left(\frac{1}{3} \ln 2\right) \cdot 9 = -3 \ln 2 \]

4. **Rewrite the equation**:
\[ n = 200 e^{-3 \ln 2} \]

5. **Use the property of exponents**:
\[ e^{-3 \ln 2} = e^{\ln(2^{-3})} = 2^{-3} = \frac{1}{8} \]

6. **Substitute back into the equation**:
\[ n = 200 \cdot \frac{1}{8} = 25 \]

7. **Calculate the number of decayed nuclei**:
\[ \text{Number of decayed nuclei} = n_0 - n = 200 - 25 = 175 \]

### Final Answer:
The number of nuclei that decayed during the first 9 seconds is **175**.