Okay, let's tackle this question. So the user is asking about bacterial growth, specifically exponential growth. The question is, if a bacterium divides every 20 minutes, starting with one, how many will there be after 3 hours?

First, I need to recall the formula for exponential growth. Bacteria reproduce by binary fission, so each division doubles the population. The formula is N = N0 * 2^(t/g), where N0 is the initial number, t is time, and g is the generation time.

Given that the generation time is 20 minutes, and the total time is 3 hours. Let me convert 3 hours to minutes to match the units. 3 hours is 180 minutes. Then, the number of generations would be 180 / 20 = 9 generations. So each generation doubles the population. Starting with 1, after 9 doublings, it's 2^9. 2^10 is 1024, so 2^9 is 512. Therefore, the answer should be 512.

Now, the options are not given, but the correct answer according to the user is option C. So I need to make sure that the explanation aligns with that. Let me check if there's another way to approach this. Maybe using the formula N = N0 * e^(rt), but since the growth is in discrete generations, the first formula is more appropriate here. The exponential growth formula with base 2 is suitable for binary fission.

Possible mistakes here could be miscalculating the number of generations. For example, if someone thought 3 hours is 3*60=180 minutes, divided by 20 gives 9, correct. Then 2^9 is 512. Another mistake could be thinking that each generation takes longer, like 30 minutes instead of 20, which would give a different result. Also, confusing exponential with linear growth, but the question specifies exponential. So the key is understanding that each division doubles the population, not adding a fixed number each time.

So the core concept here is exponential growth in bacterial populations due to binary fission. The explanation should clarify that each division doubles the population, leading to 2^n where n is the number of generations. The key steps are converting time units, calculating the number of generations, and applying the exponential formula.

The clinical pearl here is to remember that bacterial populations grow exponentially, which is critical in understanding infections and antibiotic resistance. Also, the formula is essential for microbiology and pharmacology exams where drug concentration or bacterial load is a factor.

**Core Concept**
Exponential growth in bacterial populations follows the formula *N = N₀ × 2ⁿ*, where *N₀* is the initial population, and *n* is the number of generations. Bacterial doubling time (here, 20 minutes) dictates the rate of exponential increase.

**Why the Correct Answer is Right**
The bacterium divides every 20 minutes, so in 3 hours (180 minutes), the number of divisions is 180 ÷ 20 = **9 generations**. Starting with 1 bacterium, the population after 9 doublings is *2⁹ = 512*. This follows binary fission,

✓ Correct Answer: C. 512