Given:
- Half-life of the radioactive element = 12.5 hours
- Initial quantity of the element = 256g
- Required quantity of the element = 1g

To Find:
After how much time will the quantity of the element remain 1g?

Formula:
The formula to calculate the remaining quantity of a radioactive element after a certain time is given by:
Remaining quantity = Initial quantity * (1/2)^(time/half-life)

Calculations:
Let's calculate the time it takes for the quantity to reduce from 256g to 1g.

1. Calculate the number of half-lives needed to reduce the quantity from 256g to 1g.
Quantity after n half-lives = Initial quantity * (1/2)^n
1g = 256g * (1/2)^n

2. Solve for n:
(1/2)^n = 1/256
Taking the logarithm of both sides:
n * log(1/2) = log(1/256)
n * (-0.301) = -2
n = -2 / -0.301
n ≈ 6.64

Therefore, it takes approximately 6.64 half-lives to reduce the quantity from 256g to 1g.

3. Calculate the time required for 6.64 half-lives:
Time = Number of half-lives * Half-life
Time = 6.64 * 12.5 hours
Time ≈ 83 hours

Answer:
After approximately 83 hours, the quantity of the radioactive element will remain 1g.

Explanation:
The half-life of a radioactive element is the time it takes for the quantity of the element to reduce by half. In this case, the half-life is 12.5 hours.

Using the formula for radioactive decay, we can calculate the remaining quantity of the element after a certain time. By setting the remaining quantity equal to 1g and solving for the number of half-lives, we find that it takes approximately 6.64 half-lives to reduce the quantity from 256g to 1g.

Finally, multiplying the number of half-lives by the half-life of the element gives us the total time required for the quantity to reduce to 1g. Therefore, the answer is approximately 83 hours.