If initially there were X atoms of a radioactive isotope and now there are X/16 atoms, how many half-lives have elapsed?

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Multiple Choice

If initially there were X atoms of a radioactive isotope and now there are X/16 atoms, how many half-lives have elapsed?

Explanation:
Each half-life halves the number of radioactive atoms. After n half-lives you have X × (1/2)^n. You’re told the current amount is X/16, so set X × (1/2)^n = X/16. Dividing by X gives (1/2)^n = 1/16. Since 1/16 = (1/2)^4, n = 4. So four half-lives have elapsed. Context: after one half-life you’d have X/2, after two: X/4, after three: X/8, and after four: X/16.

Each half-life halves the number of radioactive atoms. After n half-lives you have X × (1/2)^n. You’re told the current amount is X/16, so set X × (1/2)^n = X/16. Dividing by X gives (1/2)^n = 1/16. Since 1/16 = (1/2)^4, n = 4. So four half-lives have elapsed. Context: after one half-life you’d have X/2, after two: X/4, after three: X/8, and after four: X/16.

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