One-fourth of the material remaining refers to the fraction of the radioactive material that is left after a certain amount of time has passed. To find the time at which one-fourth of the material remains, we need to consider the decay of both particles with their respective half-lives.

Let's denote the initial amount of the radioactive material as N0. After one half-life of the first particle (with a half-life of 1620 years), the remaining amount of the material is N0/2. Similarly, after one half-life of the second particle (with a half-life of 810 years), the remaining amount of the material is N0/2.

To find the time at which one-fourth of the material remains, we need to equate the remaining amount of the material to N0/4, which is one-fourth of the initial amount.

Let's break down the steps to find the time after which one-fourth of the material remains:

1. After one half-life of the first particle (1620 years), the remaining amount of the material is N0/2.
2. After one half-life of the second particle (810 years), the remaining amount of the material is N0/2.
3. Now, we need to find the combined time at which both particles have decayed to one-fourth of the initial amount, N0/4.

Since the first particle has a longer half-life, it takes more time for it to decay to one-fourth of the initial amount. Therefore, the combined time will be determined by the half-life of the first particle.

Let's denote the time after which one-fourth of the material remains as t. Using the equation for exponential decay, we can write:

(N0/2) * (1/2)^(t/1620) = N0/4

We can simplify this equation by canceling out N0/4 from both sides:

(1/2)^(t/1620) = 1/2

Since the base of the exponent on both sides is the same (1/2), we can equate the exponents:

t/1620 = 1

Solving for t, we get:

t = 1620

Therefore, the time after which one-fourth of the material remains is 1620 years.