A radioactive sample has `8.0xx10^(18)` active nuclei at a certain instant. How many of these nuclei will still be in the active state after two half-life `("in" xx10^(18))`?

Explanation

To solve the problem of how many active nuclei remain after two half-lives, we can follow these steps:

### Step 1: Understand the concept of half-life
The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. After one half-life, the number of active nuclei is halved.

### Step 2: Identify the initial number of active nuclei
We are given that the initial number of active nuclei (N₀) is:
\[ N₀ = 8.0 \times 10^{18} \]

### Step 3: Calculate the number of active nuclei after one half-life
After one half-life, the number of active nuclei (N₁) is:
\[ N₁ = \frac{N₀}{2} = \frac{8.0 \times 10^{18}}{2} = 4.0 \times 10^{18} \]

### Step 4: Calculate the number of active nuclei after two half-lives
After the second half-life, the number of active nuclei (N₂) is:
\[ N₂ = \frac{N₁}{2} = \frac{4.0 \times 10^{18}}{2} = 2.0 \times 10^{18} \]

### Step 5: State the final answer
Thus, the number of active nuclei remaining after two half-lives is:
\[ N₂ = 2.0 \times 10^{18} \]

### Final Answer
The answer is \( 2.0 \times 10^{18} \).

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