Two radioactive materials `X_(1)` and `X_(2)` have decay constants `10 lamda` and `lamda` respectively. If initially they have the same number of nuclei, if the ratio of the number of nuclei of `X_(1)` to that of `X_(2)` will be `1//e` after a time `n/(9lamda)`. Find the value of `n`?

Explanation

To solve the problem, we need to analyze the decay of the two radioactive materials \(X_1\) and \(X_2\) using their decay constants. Let's break it down step by step.

### Step 1: Understand the decay law
The number of nuclei \(N\) of a radioactive material at time \(t\) can be expressed using the formula:
\[ N(t) = N_0 e^{-\lambda t} \]
where \(N_0\) is the initial number of nuclei, \(\lambda\) is the decay constant, and \(t\) is the time elapsed.

### Step 2: Write the equations for both materials
For material \(X_1\) with decay constant \(10\lambda\):
\[ N_1(t) = N_0 e^{-10\lambda t} \]
For material \(X_2\) with decay constant \(\lambda\):
\[ N_2(t) = N_0 e^{-\lambda t} \]

### Step 3: Set up the ratio of the number of nuclei
We need to find the ratio of the number of nuclei of \(X_1\) to that of \(X_2\):
\[ \frac{N_1(t)}{N_2(t)} = \frac{N_0 e^{-10\lambda t}}{N_0 e^{-\lambda t}} = e^{-10\lambda t + \lambda t} = e^{-9\lambda t} \]

### Step 4: Use the given condition
According to the problem, this ratio equals \(\frac{1}{e}\) after a time of \(\frac{n}{9\lambda}\):
\[ e^{-9\lambda t} = \frac{1}{e} \]
This implies:
\[ -9\lambda t = -1 \]
or
\[ 9\lambda t = 1 \]

### Step 5: Substitute \(t\) with \(\frac{n}{9\lambda}\)
Now, substitute \(t\) with \(\frac{n}{9\lambda}\):
\[ 9\lambda \left(\frac{n}{9\lambda}\right) = 1 \]
This simplifies to:
\[ n = 1 \]

### Final Answer
Thus, the value of \(n\) is:
\[ \boxed{1} \]