Dubnium-261 has a half-life of 27 seconds and rutherfordium-261 has a half-life of 81 seconds.Estimate what fraction of the dubnium-261 isotope remains in the same amount of time that 34 of rutherfordium-261 decays.

To solve this problem, we first need to understand the concept of half-life. The half-life of a radioactive substance is the time it takes for half of the substance to decay.

Step 1: Determine the time it takes for 3/4 of Rutherfordium-261 to decay. Since the half-life of Rutherfordium-261 is 81 seconds, this means that half of it will decay in 81 seconds. To find out when 3/4 of it has decayed, we need to find the time it takes for half of the remaining half (or 1/4) to decay. This is another half-life, or another 81 seconds. So, the total time for 3/4 of Rutherfordium-261 to decay is 81 seconds + 81 seconds = 162 seconds.

Step 2: Determine how much Dubnium-261 remains after this time. The half-life of Dubnium-261 is 27 seconds. This means that half of it will decay every 27 seconds. To find out how much remains after 162 seconds, we divide 162 by 27 to find out how many half-lives have passed. 162 ÷ 27 = 6. This means that 6 half-lives have passed.

In one half-life, half of the substance remains. In two half-lives, half of the remaining half (or 1/4) remains. In three half-lives, half of the remaining 1/4 (or 1/8) remains, and so on.

So, after 6 half-lives, the fraction of Dubnium-261 that remains is 1/2^6 = 1/64.

Therefore, after the same amount of time that 3/4 of Rutherfordium-261 decays, 1/64 of the Dubnium-261 isotope remains.