The half-life periof of a radioactive substance is `x` years. The fraction remaining after `2x` years is………….. .

Answer

To solve the problem, we need to determine the fraction of a radioactive substance that remains after a time period of \(2x\) years, given that its half-life is \(x\) years.

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life:**
The half-life (\(T_{1/2}\)) of a radioactive substance is the time required for half of the substance to decay. Given that the half-life is \(x\) years, after \(x\) years, half of the substance will remain.

2. **Calculate Remaining Fraction After One Half-Life:**
After \(x\) years, the fraction remaining is:
\[ \text{Fraction remaining after } x \text{ years} = \frac{1}{2} \]

3. **Calculate Remaining Fraction After Two Half-Lives:**
After another \(x\) years (which totals \(2x\) years), we apply the half-life concept again. The remaining fraction after another half-life is:
\[ \text{Fraction remaining after } 2x \text{ years} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

4. **Conclusion:**
Therefore, the fraction of the radioactive substance remaining after \(2x\) years is:
\[ \frac{1}{4} \]

### Final Answer:
The fraction remaining after \(2x\) years is \(\frac{1}{4}\).