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 formula for radioactive decay, which is given by:

\[ N(t) = N_0 e^{-\lambda t} \]

Where:
- \( N(t) \) is the number of nuclei remaining at time \( t \),
- \( N_0 \) is the initial number of nuclei,
- \( \lambda \) is the decay constant,
- \( t \) is the time elapsed.

### Step 1: Identify the given values
- Initial number of nuclei, \( N_0 = 200 \)
- Decay constant, \( \lambda = \frac{1}{3} \ln(2) \)
- Time, \( t = 9 \) seconds

### Step 2: Calculate the remaining nuclei after 9 seconds
Using the decay formula:

\[ N(t) = N_0 e^{-\lambda t} \]

Substituting the known values:

\[ N(9) = 200 e^{-\left(\frac{1}{3} \ln(2)\right) \cdot 9} \]

### Step 3: Simplify the exponent
We can simplify the exponent:

\[ N(9) = 200 e^{-\left(3 \ln(2)\right)} \]

Using the property of exponents:

\[ e^{-\left(3 \ln(2)\right)} = e^{\ln(2^{-3})} = 2^{-3} = \frac{1}{8} \]

### Step 4: Substitute back into the equation
Now substituting back:

\[ N(9) = 200 \cdot \frac{1}{8} \]

Calculating this gives:

\[ N(9) = 25 \]

### Step 5: Calculate the number of nuclei decayed
The number of nuclei that have decayed is given by:

\[ \text{Nuclei decayed} = N_0 - N(t) \]

Substituting the values:

\[ \text{Nuclei decayed} = 200 - 25 = 175 \]

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

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