Calculating Half-Life of Rock Element
To calculate the half-life of a rock element, we can use the formula:
Half-Life = ln(2) / Decay Constant
where, ln(2) is the natural logarithm of 2 and Decay Constant is a constant which depends on the element.

For an old rock element with a half-life of 5800 years, the decay constant can be calculated as:
Decay Constant = ln(2) / Half-Life
Decay Constant = ln(2) / 5800
Decay Constant = 0.000119

Calculating Time Taken to Reduce Sample
To calculate the time taken for a sample of 25 gm to reduce to 6.25 gm, we can use the formula:
N(t) = N(0) * e^(-λt)
where, N(t) is the amount of the element at time t, N(0) is the initial amount of the element, λ is the decay constant, and e is the mathematical constant.

Substituting the given values, we get:
6.25 = 25 * e^(-0.000119t)

Dividing both sides by 25, we get:
0.25 = e^(-0.000119t)

Taking natural logarithms on both sides, we get:
ln(0.25) = -0.000119t

Solving for t, we get:
t = ln(0.25) / -0.000119
t = 14417.69 years

Therefore, it takes 14417.69 years for a sample of 25 gm of the old rock element to reduce to 6.25 gm.

Explanation
The half-life of a rock element is the time taken for half of the element to decay. In this case, the half-life of the old rock element is 5800 years. This means that after every 5800 years, the amount of the element reduces to half of its initial value.

The decay constant is a constant which depends on the element and can be calculated using the formula. In this case, the decay constant is 0.000119.

To calculate the time taken for a sample of 25 gm to reduce to 6.25 gm, we can use the exponential decay formula. By substituting the given values, we can solve for the time taken. The answer is expressed in years.

Therefore, the time taken for the sample to reduce can be calculated using the formula and the values provided.