Final answer:
It would take approximately 13.38 billion years for 7/8 of a sample of uranium-238 to decay, based on its half-life of 4.46 × 10⁹ years.
Step-by-step explanation:
To calculate how long it would take for 7/8 of a sample of uranium-238 (U-238) to decay, we examine the half-life, which is the time required for half the atoms in a sample to decay. Uranium-238 has a half-life of 4.46 × 10⁹ years. For 7/8 of the sample to decay, we must go through three half-lives: after one half-life, 1/2 remains; after two half-lives, 1/4 remains; and after three half-lives, 1/8 remains. Therefore, it would take three full half-lives for 7/8 to decay, which is 3 × 4.46 × 10⁹ years.
Performing the calculation: (3 × 4.46 × 10⁹) years = (13.38 × 10⁹) years. It would take approximately 13.38 billion years for 7/8 of a uranium-238 sample to decay through alpha decay.